Abouzaid proved in \cite{Abouzaid-fiber-generation} that the wrapped Fukaya category of a cotangent bundle is generated by one cotangent fiber. In the filtered case, this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagrangian only when they geometrically coincide. However the filtered Fukaya category comes with a notion of interleaving distance and Paul Biran~\cite{Biran-V63}asked whether the iterated cones on the cotangent fibers generate a dense subcategory, in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent fibers. We refer to \cite{Ambrosioni-Filtered-Fukaya} for a construction of the filtered Fukaya category and~\cite{Biran-Cornea-Zhang} for its persistence structure, which yields a notion of distance. Similar density considerations had been evoked in \cite{Fukaya-GHdistance}.

We shall use the filtered Fukaya category constructed in~\cite{Ambrosioni-Filtered-Fukaya}. Its objects are the compact Lagrangians so it does not contain the cotangent fibers. To make sense of the question of density of iterated cones of the fibers, we shall embed this category in its module category through the Yoneda embedding.

Let us introduce some notations. Let $N$ be a connected closed manifold and $DT^*N$ its unit ball cotangent bundle (with respect to a Riemannian metric) with Liouville form $\lambda$.  We denote by $\Fuk(DT^*N)$ the filtered Fukaya category whose objects are compact (exact) Lagrangian branes, that is pairs $(L, f_L)$ where $L$ is a closed exact Lagrangian contained in $DT^*N$ and $f_L$ is a primitive of $\lambda_{\mid L}$. Morphisms from $L$ to $L'$ are given by the Floer cochains $FC^*(L,L')$. The space of Floer cochains is filtered once we have primitives $f_L, f_{L'}$ of $\lambda_{\mid L}, \lambda_{\mid L'}$: an intersection point $x\in L\cap L'$ has filtration degree $f_{L'}(x)-f_L(x)$.

We let $\Yo\colon \Fuk(DT^*N) \to \fMod(\Fuk(DT^*N))$ be the Yoneda embedding.  Here $\fMod(\mathcal C)$ is the set of functors from $\mathcal C$ to the category of filtered chain complexes, and the Yoneda morphism is given by associating to $L$ the functor $L' \mapsto FC^*(L',L;t)$ where $FC^*(L',L;t)$ is the set of Floer chains of action greater than $t$. For simplicity we denote by $FC^*(L,L')$ the Floer chain complex with its filtration being understood, so the Yoneda embedding will send $L$ to $L' \mapsto FC^*(L',L)$.

Let $V_{(x,a)}$ be the fiber at $x$ with primitive $a$ of $\lambda_{\mid V_x}=0$. This Lagrangian brane $V_{(x,a)}$ does not belong to $\Fuk(DT^*N)$ but it makes sense to define the filtered complex $FC^*(L', V_{(x,a)})$ for any $L' \in \Fuk(DT^*N)$ and this gives a module $\V_{(x,a)} \in \fMod(\Fuk(DT^*N))$.  The action of $FC^*(L,L')$ on $FC^*(L', V_{(x,a)})$ is given by the standard triangle product
\[FC^*(L,L')\otimes FC^*(L', V_{(x,a)}) \longrightarrow FC^*(L, V_{(x,a)})\]
which respects the filtration by \cite{Ambrosioni-Filtered-Fukaya}.

The module category $\fMod(\Fuk(DT^*N))$ inherits an interleaving distance denoted by $\gamma$.  It is also a pre-triangulated category.

We define the subcategory $\langle \V_{(x_1,a_1)}, \ldots, \V_{(x_l,a_l)} \rangle$ of $\fMod(\Fuk(DT^*N))$ the subcategory generated by the $\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.

Now we can state our main result.

Abouzaid proved in \cite{Abouzaid-fiber-generation} that the wrapped Fukaya category of a cotangent bundle is generated by one cotangent fiber. In the filtered case, this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagrangian only when they geometrically coincide. However the filtered Fukaya category comes with a notion of interleaving distance and Paul Biran~\cite{Biran-V63}asked whether the iterated cones on the cotangent fibers generate a dense subcategory, in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent fibers. We refer to \cite{Ambrosioni-Filtered-Fukaya} for a construction of the filtered Fukaya category and~\cite{Biran-Cornea-Zhang} for its persistence structure, which yields a notion of distance. Similar density considerations had been evoked in \cite{Fukaya-GHdistance}.

Let us introduce some notations. Let $N$ be a connected closed manifold and $DT^*N$ its unit ball cotangent bundle (with respect to a Riemannian metric) with Liouville form $\lambda$. We denote by $\Fuk(DT^*N)$ the filtered Fukaya category whose objects are compact (exact) Lagrangian branes, that is pairs $(L, f_L)$ where $L$ is a closed exact Lagrangian contained in $DT^*N$ and $f_L$ is a primitive of $\lambda_{\mid L}$. Morphisms from $L$ to $L'$ are given by the Floer cochains $FC^*(L,L')$. The space of Floer cochains is filtered once we have primitives $f_L, f_{L'}$ of $\lambda_{\mid L}, \lambda_{\mid L'}$: an intersection point $x\in L\cap L'$ has filtration degree $f_{L'}(x)-f_L(x)$.

We let $\Yo\colon \Fuk(DT^*N) \to \fMod(\Fuk(DT^*N))$ be the Yoneda embedding. Here $\fMod(\mathcal C)$ is the set of functors from $\mathcal C$ to the category of filtered chain complexes, and the Yoneda morphism is given by associating to $L$ the functor $L' \mapsto FC^*(L',L;t)$ where $FC^*(L',L;t)$ is the set of Floer chains of action greater than $t$. For simplicity we denote by $FC^*(L,L')$ the Floer chain complex with its filtration being understood, so the Yoneda embedding will send $L$ to $L' \mapsto FC^*(L',L)$.

Let $V_{(x,a)}$ be the fiber at $x$ with primitive $a$ of $\lambda_{\mid V_x}=0$. This Lagrangian brane $V_{(x,a)}$ does not belong to $\Fuk(DT^*N)$ but it makes sense to define the filtered complex $FC^*(L', V_{(x,a)})$ for any $L' \in \Fuk(DT^*N)$ and this gives a module $\V_{(x,a)} \in \fMod(\Fuk(DT^*N))$. The action of $FC^*(L,L')$ on $FC^*(L', V_{(x,a)})$ is given by the standard triangle product
\[FC^*(L,L')\otimes FC^*(L', V_{(x,a)}) \longrightarrow FC^*(L, V_{(x,a)})\]
which respects the filtration by \cite{Ambrosioni-Filtered-Fukaya}.

We define the subcategory $\langle \V_{(x_1,a_1)}, \ldots, \V_{(x_l,a_l)} \rangle$ of $\fMod(\Fuk(DT^*N))$ the subcategory generated by the $\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.

Now we can state our main result.

We define the subcategory $\langle \V_{(x_1,a_1)}, \ldots, \V_{(x_l,a_l)} \rangle$ of $\fMod(\Fuk(DT^*N))$ the subcategory generated by the $\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.

Now we can state our main result.

The strategy of proof is to use the quantization functor constructed in~\cite{Viterbo-Sheaves} which associates a sheaf $Q(L)$ on $N\times\Real$ with any closed exact Lagrangian brane $L$. In this way we embed $\Fuk(DT^*N)$ into $\Sh(N\times\Real)$, more precisely into the subcategory $\Sh_{DT^*N}(N\times\Real)$ (actually the Tamarkin category $\Tam_{DT^*N}(N)$ introduced later), formed by the sheaves with reduced microsupport contained in $DT^*N$. The category $\Sh_{DT^*N}(N\times\Real)$ should be understood as a model of the wrapped Fukaya category in the filtered setting. The fiber $\V_{(x,a)}$ corresponds to the sheaf $k_{\{x\} \times [a, \infty)}\in \Sh(N\times\Real)$, which does not belong to $\Sh_{DT^*N}(N\times\Real)$. We let $P'_{DT^*N}$ be a projector which takes values in $\Sh_{DT^*N}(N\times\Real)$ (it will be recalled in \S\ref{sec:strategyproof} - see~\cite{Kuo-wrappedsheaves} where it is built as a wrapping functor) and we set $\W_{(x,a)} = P'_{DT^*N}(k_{\{x\} \times [a, \infty)})$. A slight modification of~\cite[Thm. 4.4]{Zhang-Cyclic} shows that the endomorphism algebra of $\W_{(x,a)}$ computes the homology of length-filtered based loop spaces, which further clarifies the role of $\Sh_{DT^*N}(N\times\Real)$ as ``wrapped filtered Fukaya category''. The sheaf category also inherits an interleaving distance that we denote by $\gamma^s $.

Now in the category $\Sh_{DT^*N}(N\times\Real)$ we can prove a more general result:
\begin{thm}[Density Theorem]\label{thm:density sheaf}
 Let $\F \in \Sh_{DT^*N}(N\times\Real)$ such that for all $x\in N$ the sheaf $\F \otimes k_{\{x\} \times\Real}$ is a $\gamma^s$-limit of constructible sheaves on $\Real$. Then, for any $\varepsilon >0$, there are points $(x_i)_{i\in \{1, \ldots,l\}}$, real numbers $(a_i)_{i\in \{1, \ldots,l\}}$ and $C \in \langle \W_{(x_1,a_1)}, \ldots, \W_{(x_l,a_l)} \rangle$ in ${\Sh_{DT^*N}(N\times\Real)}$ such that $\gamma^s (\F,C) < \varepsilon$.
\end{thm}

\begin{rem}
In fact, as in \cite{A-B-C} we prove something stronger than density, we prove what they call
{\it approximability}: For any $\varepsilon>0$ we can find objects $X_1,...,X_N$ such that any object in the category is at distance at most $ \varepsilon$ from an iterated cone of the $X_k$. Even better if we consider the set of all direct sums obtained from the $X_i$, we need only $n+1$-iterated cones (where $n$ is the dimension of the base manifold). A related notion of complexity called the interleaving Rouquier dimension will be discussed in Appendix~\ref{sec:InterleavingRouquierdimension}.
\end{rem}
Once we have the result for sheaves, we can come back to $\fMod(\Fuk(DT^*N))$ using the quantization functor $Q$ since sheaf quantization of closed exact Lagrangian satisfies the assumption of the Density Theorem, due to constructibility of $Q(L) \otimes k_{\{x\} \times\Real} \simeq FC^*(V_{(x,0)},L)$ as sheaves on $\Real$. In fact, the sheaf result is a little stronger in the sense that it may apply to sheaves associated with immersed Lagrangians or certain $C^0$-Lagrangians. One should also notice that the Fukaya category may depend on perturbation data; however, the sheaf category does not. Then it tells that the sheaf distance bound gives a uniform bound of Fukaya category distance for all perturbation data.

\begin{thm}[Density Theorem for sheaves]\label{thm:density sheaves}
Let $\F \in \Tam_{DT^*N}(N\times\Real)$ such that for any $x\in N$ the sheaf $\F \otimes k_{\{x\} \times\Real}$ is a $\gamma$-limit of constructible sheaves on $\Real$. Then for any $\varepsilon >0$, there are points $(x_i)_{i\in \{1, \ldots,l\}}$, real numbers $(a_i)_{i\in \{1, \ldots,l\}}$ and $C \in \langle \W_{(x_1,a_1)}, \ldots, \W_{(x_l,a_l)} \rangle$ in ${\Tam_{DT^*N}(N\times\Real)}$ such that $\gamma^s (\F,C) < \varepsilon$.
\end{thm}

\begin{lem}\label{lem:approx fibre boule}
 Let $x\in N$, $\varepsilon >0$ and let $Z\subset N$ be a closed contractible subset contained in the ball with center $x$ and radius $\varepsilon < r_{inj}(N)/3$. Then, for any $\F \in \Sh(N\times\Real)$ such that $SS(\F) \subset \{\tau \geq \| p \| \}$, we have $\gamma_{h^+}(\F \otimes k_{\{x\} \times \Real}, \F \otimes k_{Z \times \Real}) \leq 4\varepsilon$.
\end{lem}
\begin{proof}
 We recall that $h^+(x,p,t,\tau) = \max\{ \tau, \| p \| \}$ for $\tau \geq 0$. Since we work on $\Sh_{\tau\geq0}(N\times\Real)$ we can as well assume $h^+(x,p,t,\tau) = \max\{ \tau, \| p \| \}$ for any $\tau$. We first give a bound on $N^2\times \Real^2$ and then take the pull-back by the diagonal embedding $\delta \colon N\times\Real \to (N\times\Real)^2$. To distinguish, we denote points of the second copy of $T^*(N\times\Real)$ with prime, and then points of $T^*(N^2\times\Real^2)$ are denoted by $(x,p,x',p',t,\tau, t',\tau')$. We will apply Lemma~\ref{lem:distance restriction} and we look for a function $h_1$ on $T^*(N^2\times\Real^2)$ such that $h_1|_{\Delta_1} = h^+ \circ (d\delta)^t$, where $\Delta_1 = \Delta_{N\times\Real} \times_{N^2\times\Real^2} T^*(N^2\times\Real^2)$. We have $(d\delta)^t(x,p,x,p',t,\tau, t,\tau') = (x,t, p+p', \tau+\tau')$ so we should extend the function $\|p+p'\|$ outside the diagonal. We let $\Delta(r)$ be the neighborhood of $\Delta_N$ formed by the pairs $(x,x')$ such that $d(x,x') < r$. For $(x,x') \in \Delta(r_{inj}(N))$ we identify $T_x^*N$ and $T_{x'}^*N$ through the parallel transport along the shortest geodesic between $x$ and $x'$; we can then make sense of $p+p'$. We choose a partition of unity $\alpha, \beta$ with $\alpha=1$ on $\Delta(r_{inj}(N)/2)$ and $\beta=1$ outside $\Delta(r_{inj}(N))$. We set $h_0(x,p,x,p',t,\tau, t,\tau') = \alpha \|p+p'\| + \beta \|(p,p')\|$. We have the inequality
\[
 \|p\| + h_0 \geq \|p'\|
 \]
 because over $\Delta(r_{inj}(N))$ we have $\|p\| + h_0 \geq \alpha \|p\| + \alpha \|p+p'\| + \beta \|p'\| \geq (\alpha + \beta)\|p'\| = \|p'\|$ and outside $\Delta(r_{inj}(N))$ we even have $h_0 \geq \|p'\|$.

\subsection{Proof of Theorem~\ref{thm:density sheaves}}\label{sec:proofmainthm}
Let $0<\varepsilon < r_{inj}(N)/3$. We pick a finite covering $N = \bigcup_{i\in I} U_i$ be a finite covering as in Lemma \ref{lem:Cech} and assume moreover that all $U_i$'s are contained in a ball of radius less than $\varepsilon$. We set $U_J^{cl} = \bigcap_{j\in J} \overline{U_j}$. By Lemma \ref{lem:Cech} $\F$ is an $n$ steps iterated cone on $\F \otimes C_0[1]$, $\F \otimes C_1$, \dots, $\F \otimes C_n[1-n]$, where $C_i = \bigoplus_{J \subset I, |J| = i+1} k_{U_J^{cl}\times \Real}$. For each $J\subset I$ we pick $x_J$ such that $U_J^{cl} \subset B_\varepsilon(x_J )$. Lemma \ref{lem:approx fibre boule} says that $\gamma_{h^+}(\F \otimes k_{\{x_J\} \times \Real}, \F \otimes k_{U_J^{cl} \times \Real}) \leq 4\varepsilon$. We set $C'_i = \bigoplus_{J \subset I, |J| = i+1} k_{\{x_J\}\times \Real}$. By Lemma~\ref{lem:distance somme} we have $\gamma_{h^+}(\F \otimes C_i, \F \otimes C'_i) \leq 8 \varepsilon$.

The strategy of proof is to use the quantization functor constructed in~\cite{Viterbo-Sheaves} which associates a sheaf $Q(L)$ on $N\times{\mathbb R}$ with any closed exact Lagrangian brane $L$.  In this way we embed $\Fuk(DT^*N)$ into $\Sh(N\times{\mathbb R})$, more precisely into the subcategory $\Sh_{DT^*N}(N\times{\mathbb R})$ (actually the Tamarkin category $\Tam_{DT^*N}(N)$ introduced later), formed by the sheaves with reduced microsupport contained in $DT^*N$. The category $\Sh_{DT^*N}(N\times{\mathbb R})$ should be understood as a model of the wrapped Fukaya category in the filtered setting. The fiber $\V_{(x,a)}$ corresponds to the sheaf $k_{\{x\} \times [a, \infty)}\in \Sh(N\times{\mathbb R})$, which does not belong to $\Sh_{DT^*N}(N\times{\mathbb R})$. We let $P'_{DT^*N}$ be a projector which takes values in $\Sh_{DT^*N}(N\times{\mathbb R})$ (it will be recalled in \S\ref{sec:strategyproof} - see~\cite{Kuo-wrappedsheaves} where it is built as a wrapping functor) and we set $\W_{(x,a)} = P'_{DT^*N}(k_{\{x\} \times [a, \infty)})$. A slight modification of~\cite[Thm. 4.4]{Zhang-Cyclic} shows that the endomorphism algebra of $\W_{(x,a)}$ computes the homology of length-filtered based loop spaces, which further clarifies the role of $\Sh_{DT^*N}(N\times{\mathbb R})$ as ``wrapped filtered Fukaya category''. The sheaf category also inherits an interleaving distance that we denote by $\gamma^s $.

Now in the category $\Sh_{DT^*N}(N\times{\mathbb R})$ we can prove a more general result:
\begin{thm}[Density Theorem]\label{thm:density sheaf}
  Let ${\mathcal F} \in \Sh_{DT^*N}(N\times{\mathbb R})$ such that for all $x\in N$ the sheaf ${\mathcal F} \otimes k_{\{x\} \times{\mathbb R}}$ is a $\gamma^s$-limit of constructible sheaves on ${\mathbb R}$. Then, for any $\varepsilon >0$, there are points $(x_i)_{i\in \{1, \ldots,l\}}$, real numbers $(a_i)_{i\in \{1, \ldots,l\}}$ and $C \in \langle \W_{(x_1,a_1)}, \ldots, \W_{(x_l,a_l)} \rangle$ in ${\Sh_{DT^*N}(N\times{\mathbb R})}$ such that $\gamma^s ({\mathcal F},C) < \varepsilon$.
\end{thm}

The idea is to use a \v Cech resolution of the constant sheaf $k_N \simeq C_0 \to C_1 \to \cdots \to C_m$, where $C_i = \bigoplus_{J \subset I, |J| = i+1} k_{U_J^{cl}}$ with $U_J^{cl} = \bigcap_{j\in J} \overline{U_j}$ and the $U_j$'s are $\varepsilon$-small balls covering $N$. Then ${\mathcal F} = {\mathcal F} \otimes k_{N\times{\mathbb R}}$ is written as an iterated cone on the ${\mathcal F} \otimes k_{U_J^{cl} \times{\mathbb R}}$. Now ${\mathcal F} \otimes k_{U_J^{cl} \times{\mathbb R}}$ can be approximated by ${\mathcal F} \otimes k_{\{x\} \times{\mathbb R}}$ for some $x \in U_J^{cl}$ and ${\mathcal F} \otimes k_{\{x\} \times{\mathbb R}} $ (a sheaf on a line) can be written as (in general can be approximated by) an iterated cone on the $\W_{(x,a)}$'s by constructibility. From the argument, we know that the points $(x_i)_{i\in \{1, \ldots,l\}}$ can be taken from an a priori given dense subset of the base manifold $N$. Also, the proof may be viewed as a filtered analogue of the sectorial descent of~\cite{G-P-S-3}, formulated in the language of sheaves.

\begin{rem}
In fact, as in \cite{A-B-C} we prove something stronger than density, we prove what they call
{\it approximability}: For any $\varepsilon>0$ we can find objects $X_1,...,X_N$ such that any object in the category is at distance at most $ \varepsilon$ from an iterated cone of the $X_k$. Even better if we consider the set of all direct sums  obtained from the $X_i$, we need only $n+1$-iterated cones (where $n$ is the dimension of the base manifold). A related notion of complexity called the interleaving Rouquier dimension will be discussed in Appendix~\ref{sec:InterleavingRouquierdimension}.
\end{rem}
Once we have the result for sheaves, we can come back to $\fMod(\Fuk(DT^*N))$ using the quantization functor $Q$ since sheaf quantization of closed exact Lagrangian satisfies the assumption of the Density Theorem, due to constructibility of $Q(L) \otimes k_{\{x\} \times{\mathbb R}} \simeq FC^*(V_{(x,0)},L)$ as sheaves on ${\mathbb R}$. In fact, the sheaf result is a little stronger in the sense that it may apply to sheaves associated with immersed Lagrangians or certain $C^0$-Lagrangians. One should also notice that the Fukaya category may depend on perturbation data; however, the sheaf category does not. Then it tells that the sheaf distance bound gives a uniform bound of Fukaya category distance for all perturbation data.

\medskip

To implement this plan of proof we need to be more precise on the quantization functor $Q$. In~\cite{Viterbo-Sheaves} the quantization functor is defined on the Donaldson-Fukaya category of $T^*N$ with value in the (classical) derived category of sheaves. However the Donaldson-Fukaya category is not triangulated and we cannot state our result in this framework. For this reason we will first enhance $Q$ to a functor defined on the Fukaya category in a higher coherent way, and then we can extent $Q$ to the category of modules over Fukaya category.  Recall that the starting point in~\cite{Viterbo-Sheaves} is to first define a presheaf whose sections on some open set $U \times (-\infty, a)$ is $\lim_f FH(\Gamma_{df}, L)$ where $f$ is a function running over the smooth functions greater than the characteristic function of $U$, rescaled by $a$.  To turn this into a functor we need some functoriality of homotopy limits. This may be true in the framework of $A_\infty$-categories but we lack references. So we will turn our categories into $\infty$-categories, for which the appropriate results are available in the literature. In the sequel and for simplicity, all coefficients will be in $\mathbb Z/2 \mathbb Z$.

To prove Theorem~\ref{thm:density Fukaya} the strategy is to use the quantization functor of~\cite{Viterbo-Sheaves}, which associates to a closed exact Lagrangian brane $L$ a sheaf $Q(L)$ on $N\times\mathbb R$, thereby embedding $\Fuk(DT^*N)$ into the sheaf category $\Sh_{DT^*N}(N\times\mathbb R)$ (the Tamarkin category). One introduces the objects $\W_{(x,a)}:=P'_{DT^*N}(k_{\{x\}\times[a,\infty)})$ using a projector $P'_{DT^*N}$ landing in $\Sh_{DT^*N}(N\times\mathbb R)$, and one proves first a sheaf-theoretic density statement (Theorem~\ref{thm:density sheaf}) with respect to the sheaf interleaving distance $\gamma^s$.

The proof idea for the sheaf density theorem is: use a \v Cech resolution of $k_N\simeq C_0\to C_1\to\cdots\to C_m$ where $C_i=\bigoplus_{J\subset I,\ |J|=i+1} k_{U_J^{cl}}$ for an $\varepsilon$-small ball cover $(U_j)$. Then write $\mathcal F=\mathcal F\otimes k_{N\times\mathbb R}$ as an iterated cone on the pieces $\mathcal F\otimes k_{U_J^{cl}\times\mathbb R}$. Next, approximate $\mathcal F\otimes k_{U_J^{cl}\times\mathbb R}$ by $\mathcal F\otimes k_{\{x\}\times\mathbb R}$ for some $x\in U_J^{cl}$, and (since this is a sheaf on a line) use constructibility to write (or approximate) $\mathcal F\otimes k_{\{x\}\times\mathbb R}$ as an iterated cone on the $\W_{(x,a)}$'s. The argument also shows the points $x_i$ can be chosen from an a priori given dense subset of $N$.

Finally, ‘once we have the result for sheaves’, one returns to $\fMod(\Fuk(DT^*N))$ via $Q$: sheaf quantization of closed exact Lagrangians satisfies the hypothesis of the sheaf density theorem because $Q(L)\otimes k_{\{x\}\times\mathbb R}\simeq FC^*(V_{(x,0)},L)$ is constructible as a sheaf on $\mathbb R$. This transfers the approximation in $\gamma^s$ to the desired density statement for $L$ in terms of the generators $\V_{(x_i,a_i)}$.

To prove the main theorem the strategy is to use the quantization functor of~\cite{Viterbo-Sheaves}, which associates to a closed exact Lagrangian brane $L$ a sheaf $Q(L)$ on $N\times\mathbb R$, thereby embedding $\Fuk(DT^*N)$ into the sheaf category $\Sh_{DT^*N}(N\times\mathbb R)$ (the Tamarkin category). One introduces the objects $\W_{(x,a)}:=P'_{DT^*N}(k_{\{x\}\times[a,\infty)})$ using a projector $P'_{DT^*N}$ landing in $\Sh_{DT^*N}(N\times\mathbb R)$, and one proves first a sheaf-theoretic density statement (Theorem~\ref{thm:density sheaf}) with respect to the sheaf interleaving distance $\gamma^s$.

The proof idea for the sheaf density theorem is: use a \v Cech resolution of $k_N\simeq C_0\to C_1\to\cdots\to C_m$ where $C_i=\bigoplus_{J\subset I,\ |J|=i+1} k_{U_J^{cl}}$ for an $\varepsilon$-small ball cover $(U_j)$. Then write $\mathcal F=\mathcal F\otimes k_{N\times\mathbb R}$ as an iterated cone on the pieces $\mathcal F\otimes k_{U_J^{cl}\times\mathbb R}$. Next, approximate $\mathcal F\otimes k_{U_J^{cl}\times\mathbb R}$ by $\mathcal F\otimes k_{\{x\}\times\mathbb R}$ for some $x\in U_J^{cl}$, and (since this is a sheaf on a line) use constructibility to write (or approximate) $\mathcal F\otimes k_{\{x\}\times\mathbb R}$ as an iterated cone on the $\W_{(x,a)}$'s. The argument also shows the points $x_i$ can be chosen from an a priori given dense subset of $N$.

Finally, once we have the result for sheaves, one returns to $\fMod(\Fuk(DT^*N))$ via $Q$: sheaf quantization of closed exact Lagrangians satisfies the hypothesis of the sheaf density theorem because $Q(L)\otimes k_{\{x\}\times\mathbb R}\simeq FC^*(V_{(x,0)},L)$ is constructible as a sheaf on $\mathbb R$. This transfers the approximation in $\gamma^s$ to the desired density statement for $L$ in terms of the generators $\V_{(x_i,a_i)}$.

[Density Theorem]\label{thm:density Fukaya}
  For any closed exact Lagrangian $L \in \Fuk(DT^*N)$ and $\varepsilon >0$, there are points $(x_i)_{i\in \{1, \ldots,l\}}$, real numbers $(a_i)_{i\in \{1, \ldots,l\}}$ and $C \in \langle \V_{(x_1,a_1)}, \ldots, \V_{(x_l,a_l)} \rangle$ such that $\gamma (L,C) < \varepsilon$.,

Universal–Existential
Inequality or Bound

Question: Let $N$ be a connected closed manifold, let $DT^*N$ be its unit ball cotangent bundle, and let $\Fuk(DT^*N)$ be the filtered Fukaya category of compact exact Lagrangian branes. For $x\in N$ and $a\in \mathbb R$, let $V_{(x,a)}$ denote the filtered module associated to the cotangent fiber at $x$ with primitive value $a$, and let $\langle V_{(x_1,a_1)},\dots,V_{(x_l,a_l)}\rangle$ be the full subcategory of filtered modules generated from these modules by iterated cones. Using the Yoneda embedding, view any $L\in \Fuk(DT^*N)$ as a filtered module, and let $\gamma$ denote the interleaving distance coming from the persistence structure on the filtered Fukaya category. Under these assumptions, which quantitative estimate holds for an arbitrary closed exact Lagrangian $L\in \Fuk(DT^*N)$?

For every closed exact Lagrangian $L\in \Fuk(DT^*N)$ and every $\varepsilon>0$, there exist finitely many points $x_1,\dots,x_l\in N$, real numbers $a_1,\dots,a_l\in \mathbb R$, and an object $C\in \langle V_{(x_1,a_1)},\dots,V_{(x_l,a_l)}\rangle$ such that $\gamma(L,C)<\varepsilon$.

Choices:
B. There exists >0 such that for every closed exact Lagrangian $L\in \Fuk(DT^*N)$, there exist finitely many points $x_1,\dots,x_l\in N$, real numbers $a_1,\dots,a_l\in \mathbb R$, and an object $C\in \langle V_{(x_1,a_1)},\dots,V_{(x_l,a_l)}\rangle$ such that $\gamma(L,C)<\delta$.
C. For every closed exact Lagrangian $L\in \Fuk(DT^*N)$, there exist finitely many points $x_1,\dots,x_l\in N$, real numbers $a_1,\dots,a_l\in \mathbb R$, and an object $C\in \langle V_{(x_1,a_1)},\dots,V_{(x_l,a_l)}\rangle$ such that $\gamma(L,C)<\infty$.
D. For every closed exact Lagrangian $L\in \Fuk(DT^*N)$ and every $\varepsilon>0$, there exist finitely many points $x_1,\dots,x_l\in N$ and a single real number $a\in \mathbb R$ such that, if one sets $a_i=a$ for all $i$, then there is an object $C\in \langle V_{(x_1,a)},\dots,V_{(x_l,a)}\rangle$ with $\gamma(L,C)<\varepsilon$.
E. For every closed exact Lagrangian $L\in \Fuk(DT^*N)$ and every $\varepsilon>0$, there exist finitely many points $x_1,\dots,x_l\in N$, real numbers $a_1,\dots,a_l\in \mathbb R$, and an object $C\in \langle V_{(x_1,a_1)},\dots,V_{(x_l,a_l)}\rangle$ such that $L$ is quasi-isomorphic to $C$ and $\gamma(L,C)<\varepsilon$.

The stem does not explicitly state the target estimate or directly reveal choice A. It introduces the setup and asks which quantitative conclusion holds, so the correct answer is not leaked verbatim from the stem.

The item is very close to a theorem-recall question: the correct option is essentially the sharp theorem statement with nearby quantifier perturbations. It is not a pure restatement because the choices vary meaningfully, but it remains only a mild reformulation.

Some reasoning is required to compare quantifiers and distinguish approximation from exact quasi-isomorphism, uniform error bounds, and shared primitive shifts. However, the task is mostly recognition of the theorem’s precise statement rather than substantial mathematical generation.

B, D, and E are plausible distractors tied to common theorem-reading errors, but C is a weaker statement that is also true if A is true. That makes the distractor set non-exclusive and weakens the MCQ’s validity.

Technically sophisticated but mostly theorem-recall. The distractors are partly well designed, yet the presence of a weaker true option (C) significantly reduces the question’s quality as a single-correct MCQ.


\setcounter{equation}{0}
\setcounter{theo}{0}
\setcounter{figure}{0}
At the conference {\it Fractal Geometry and Stochastics V} held in Tabarz in 2014, I gave a survey talk entitled `Sixty years of fractal projections', a version of which, written with Jon Fraser and Xiong Jin \cite{FFJ}, appeared in in the conference proceedings \cite{BFZ}.  This marked the sixtieth anniversary of the publication of John Marstrand's 1954 paper \cite{Mar} `Some fundamental geometrical properties of plane sets of fractional dimensions'  in the Proceedings of the London Mathematical Society.  For a long time the paper attracted little attention, but since the 1980s, Marstrand's projection theorems have become the prototype for  many results in fractal geometry with numerous variants and applications. This area is now more intensively researched than ever,  drawing on modern techniques from ergodic theory, CP processes, Fourier transforms, discretisation of problems and additive combinatorics, together with a lot of ingenuity.

My talk `Seventy years of fractal projections' at {\it Fractal Geometry and Stochastics VII} held in 2024 in Chemnitz highlighted some of the very considerable progress in the area over the past 10 years. This account also surveys some of this more recent work and is a sequel to \cite{FFJ} where many earlier results and references may be found. There are many excellent articles and books which provide other substantial overviews of projection properties, including references \cite{Fra2,Mat3,Mat4,Mat5,Mat2,Mat6,Shm}. Also, volumes of conference proceedings, in particular of the German `Fractal Geometry and Stochastics' and the French `Fractals and Related Fields' meetings, contain many enlightening surveys on this and many other aspects of fractal geometry.

\subsection{General remarks}
Most of this article concerns orthogonal projections of sets in the plane onto straight lines or from
$\mathbb{R}^n$ onto $m$-dimensional subspaces.
In the plane we let $L_\theta$ be the line making  angle $\theta \in [0,\pi)$ with the $x$-axis, and write $\proj_\theta: \mathbb{R}^2 \to L_\theta$ for orthogonal projection onto $L_\theta$, see Figure 1. By slight abuse of notation we will write ${\mathcal L} $ for Lebesgue measure on any line $L_\theta$, identified with $\mathbb{R}$ in the obvious way.

 \begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{projcol.png}

Figure 1: Projection of a set $E$ onto a line in direction $\theta$.
\end{center}
\end{figure}
In the higher dimensional setting, we write $G(n,m)$ for the Grassmanian of $m$-dimensional subspaces of $\mathbb{R}^n$, where $1\leq m <n$. Then $G(n,m)$ is an $m(n-m)$-dimensional compact manifold which carries a natural invariant measure $\gamma_{n,m}$ that is locally equivalent to $m(n-m)$-dimensional Lebesgue measure. When we write `for almost all $V\in G(n,m)$' this is taken to be with respect to $\gamma_{n,m}$. For $V\in G(n,m)$ let $\proj_V : \mathbb{R}^n\to V$ be orthogonal projection onto  the $m$-dimensional subspace $V$, writing ${\mathcal L}^m$ for $m$-dimensional Lebesgue measure on $V$ which we identify with $\mathbb{R}^m$.

We make the convention that we use $\proj_\theta$ specifically for the planar setting and   $\proj_V$ for the more general $G(n,m)$ context.

Here our main interest is in projections of sets rather than measures, though many results for projections of sets have measure analogues, indeed many proofs concerning projections of sets depend on consideration of projections of suitable measures supported by the sets. Lower and upper Hausdorff and packing dimensions of measures can be defined in terms of local dimensions, and may all be equal for dynamically defined measures. Then often the dimension of projected measures can be expressed in terms of entropies and Lyapunov exponents.

We will assume, sometimes without mentioning it specifically,  that the subsets of $\mathbb{R}^m$  with which we work  are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note  that recently Lutz and Stull \cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.

\subsection{Marstrand's projection theorem}
Marstrand's projection theorem in the plane \cite{Mar} may be stated as follows.

\subsection{General remarks}
Most of this article concerns orthogonal projections of sets in the plane onto straight lines or from
$\mathbb{R}^n$ onto $m$-dimensional subspaces.
In the plane we let $L_\theta$ be the line making angle $\theta \in [0,\pi)$ with the $x$-axis, and write $\proj_\theta: \mathbb{R}^2 \to L_\theta$ for orthogonal projection onto $L_\theta$, see Figure 1. By slight abuse of notation we will write ${\mathcal L} $ for Lebesgue measure on any line $L_\theta$, identified with $\mathbb{R}$ in the obvious way.

Figure 1: Projection of a set $E$ onto a line in direction $\theta$.
\end{center}
\end{figure}
In the higher dimensional setting, we write $G(n,m)$ for the Grassmanian of $m$-dimensional subspaces of $\mathbb{R}^n$, where $1\leq m <n$. Then $G(n,m)$ is an $m(n-m)$-dimensional compact manifold which carries a natural invariant measure $\gamma_{n,m}$ that is locally equivalent to $m(n-m)$-dimensional Lebesgue measure. When we write `for almost all $V\in G(n,m)$' this is taken to be with respect to $\gamma_{n,m}$. For $V\in G(n,m)$ let $\proj_V : \mathbb{R}^n\to V$ be orthogonal projection onto the $m$-dimensional subspace $V$, writing ${\mathcal L}^m$ for $m$-dimensional Lebesgue measure on $V$ which we identify with $\mathbb{R}^m$.

We make the convention that we use $\proj_\theta$ specifically for the planar setting and $\proj_V$ for the more general $G(n,m)$ context.

Here our main interest is in projections of sets rather than measures, though many results for projections of sets have measure analogues, indeed many proofs concerning projections of sets depend on consideration of projections of suitable measures supported by the sets. Lower and upper Hausdorff and packing dimensions of measures can be defined in terms of local dimensions, and may all be equal for dynamically defined measures. Then often the dimension of projected measures can be expressed in terms of entropies and Lyapunov exponents.

We will assume, sometimes without mentioning it specifically, that the subsets of $\mathbb{R}^m$ with which we work are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note that recently Lutz and Stull \cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.

\subsection{Marstrand's projection theorem}
Marstrand's projection theorem in the plane \cite{Mar} may be stated as follows.

We will assume, sometimes without mentioning it specifically, that the subsets of $\mathbb{R}^m$ with which we work are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note that recently Lutz and Stull \cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.

The natural higher dimensional analogue of Marstrand's theorem was first presented by Mattila \cite{Mat} in 1975.

Briefly, the potential-theoretic method depends on the characterisation of Hausdorff dimension in terms of energy integrals. We write $\mathcal{M}(E)$ for the positive finite Borel measures supported on $E\subset \mathbb{R}^n$. For $s>0$ we write:
\begin{equation}\label{endef}
I^s(\mu) := \int \!\! \int \frac{d\mu(x)d\mu(y)}{|x-y|^s}
\end{equation}
for the $s$-{\it energy} of the measure $\mu\in \mathcal{M}(E)$. Then
\begin{equation*}\label{endefhau}
\dh E = \sup\Big\{s : \mbox{ there exists $\mu \in \mathcal{M}(E)$ such that $I^s(\mu) < \infty$}\Big\}.
\end{equation*}
Theorem \ref{marthm}(i) may then be proved by noting that for all $0<s< \dh E$ there is a measure $\mu\in \mathcal{M}(E)$ such that $I^s(\mu) <\infty$, and verifying that $\int I^s(\mu_\theta) d\theta \leq c_sI^s(\mu)$, where $\mu_\theta$ is the projection of the measure $\mu$ onto the line $L_\theta$ in direction $\theta$, given by $\mu_\theta(F) = \mu(\proj^{-1}_\theta F)$ for $F\subset L_\theta$. The higher dimensional case, Theorem \ref{higherdim}(i), may be proved in a similar way.

Whilst Marstrand's Theorem gives that the Lebesgue measure of the `exceptional set' of projection directions $\theta$ for which $\dh \mbox{\rm proj}_\theta E < \min\{\dh E, 1\}$ has Lebesgue measure 0, often the set of exceptional directions must be somewhat smaller than this. Indeed Kaufman's potential theoretic approach \cite{Kau} also yields an upper bound for the dimension of the exceptional set in Theorems \ref{marthm}(i) and \ref{higherdim}(i), namely that if $E \subset \mathbb{R}^n$ and $0\leq s< \dh E < m$ then
\begin{equation}\label{Kaut}
\dh\{V \in G(n,m) : \dh \mbox{\rm proj}_V E< s\} \leq m(n-m) -(m-s).
\end{equation}
(This bound is written in this form for comparison with $m(n-m)$, the dimension of the Grassmanian $G(n,m))$.
Using Fourier transforms, Falconer \cite{Fal1} in 1982 obtained bounds for the exceptional sets of an alternative form: if $0\leq s \leq m \leq \dh E$ then
\begin{equation}\label{Falt}
\dh\{V \in G(n,m) : \dh \mbox{\rm proj}_V E< s\} \leq \max\{m(n-m) -(\dh E-s), 0\},
\end{equation}
and if $m \leq \dh E \leq n$,
\begin{equation}\label{Falt2}
\dh\{V \in G(n,m) : \mathcal{L}^n(\mbox{\rm proj}_V E) =0\} \leq m(n-m) -(\dh E-m);
\end{equation}
the idea here is the greater the `excess dimension' $ \dh E - s$ or $ \dh E - m$, the smaller the set of exceptional $V$.
Estimates for the dimensions of exceptional sets in the spirit of \eqref{Kaut}, where there is a direct estimate, and \eqref{Falt}--\eqref{Falt2}, where the bound depends on by how much the dimension of a set exceeds a threshold, have become known as `Kaufman-type' and `Falconer-type' estimates respectively.

\begin{theo}\label{assd}
Let $\log 3/\log 5 <s<1$. Then there is $\epsilon>0$ such that
${\dim}_{\rm A}\proj_\theta E_s=1$ for almost all $\theta \in (-\epsilon, \epsilon)$ and ${\dim}_{\rm A}\proj_\theta E_s=s<1$ for all $\theta \in (\pi/4 -\epsilon, \pi/4+ \epsilon)$.
\end{theo}
\noindent This example, due to Fraser and Orponen \cite{FO}, depends on the fact that a self-similar $E\subset \mathbb{R}^2$ defined by contracting homotheties has
$$
{\dim}_{\rm A}\mbox{proj}_\theta E =
\left\{
\begin{array}{ll}
 \dh \proj_\theta E& \text{ if } \mathcal{H}^{\dh {\rm proj}_\theta E}(\proj_\theta E) >0 \\
 1 & \text{ if }\mathcal{H}^{\dh {\rm proj}_\theta E}(\proj_\theta E)=0
\end{array}
\right.,
$$
where $\mathcal{H}^s$ is $s$-dimensional Hausdorff measure, so the construction is equivalent to finding a set of homotheties for which the IFS attractor $E$ has projections with positive or zero ($\proj_\theta E$)-dimensional Hausdorff measure in the appropriate directions.

\begin{theo}\label{projgeneral}
Let $P \subset \mathbb{R}^k$ be an open parameter set and let $E \subset \mathbb{R}^n$ be a Borel or analytic set. Let $\{V(t)\subset G(n,m): t \in P\}$ be a family of subspaces such that $V$ is $C^1$ with the derivative $D_t V(t)$ injective for all $t \in P$. Then, for all $l=0,1,\ldots,m$ and ${\mathcal L}^k$-almost all $t \in P$,
$$
\dh\proj_{V(t)} E \geq
\left\{
\begin{array}{llrl}
 \dh E-p(l) &\ \mbox{if} & p(l) + l & \leq \dh E \leq p(l) + l +1 \\
l+1 &\ \mbox{if} & \ p(l) + l +1 &\leq \dh E \leq p(l+1) + l +1
\end{array}
\right. .
$$
Moreover, if $\dh E > p(m-1) + m$ then ${\mathcal L}^m ( \proj_{V(t)} E) > 0$ for ${\mathcal L}^k$-almost all $t \in P$.
\end{theo}

\begin{theo}\label{projline}
Let $E \subset \mathbb{R}^3$ be a Borel or analytic set and let $\theta: [0,1] \to S^2$ be a non-degenerate family of directions. Then
\begin{equation}\label{projlineeq}
\dh\proj_{\theta(t)} E = \min\{\dh E, 1\}\quad \text{ for almost all } t\in [0,1],
\end{equation}
and if $\dh E >1$ then
\begin{equation}\label{resmes}
{\mathcal L} (\proj_{\theta(t)} E)>0 \quad \text{ for almost all } t\in [0,1].
\end{equation}
Concerning exceptional directions, if $0\leq s\leq \min\{\dh E, 1\}$ then
\begin{equation}\label{PYZ}
 \dh\{t : \dh \proj_{\theta(t)} E <s\} \leq s,
\end{equation}
and if $0\leq s\leq 1$ then
\begin{equation}\label{GGM}
 \dh \{t: \proj_{\theta(t)} E <s\} \leq \max\big\{0, 1 + {\textstyle \frac12} (s-\dh E) \big\},
\end{equation}
and
\begin{equation}\label{resmes1}
\dh\{t: {\mathcal L} (\proj_{\theta(t)} E) = 0 \} \leq \textstyle{\frac13}(4-\dh E).
\end{equation}
\end{theo}

\begin{theo}\label{projplane}
Let $E \subset \mathbb{R}^3$ be a Borel or analytic set and let $\theta: [0,1] \to S^2$ be a non-degenerate family of directions.
Then
\begin{equation}\label{projplaneeq}
\dh\proj_{V_{\theta(t)}} E = \min\{\dh E, 2\} \quad \text{ for almost all } t\in [0,1].
\end{equation}
Moreover, if $\dh E >2$ then
\begin{equation}\label{projlplanex}
{\mathcal L^2} (\proj_{\theta(t)} E)>0 \quad \text{ for almost all } t\in [0,1].
\end{equation}
For exceptional directions, if $0\leq s\leq 1$ then
\begin{equation}\label{GGG}
 \dh\{t : \dh \proj_{\theta(t)} E <s\} \leq \max\{ 1 + s-\dh E, 0 \}.
\end{equation}
\end{theo}

\begin{theo}\label{higherdim}
Let $E \subset \mathbb{R}^n$ be a Borel or analytic set. Then

(i) $\dh\mbox{\rm proj}_V E \leq \min\{\dh E, m\}$  with equality for $\gamma_{n,m}$-almost all $V \in G(n,m)$,
\smallskip

(ii) if $\dh E >m$ then ${\mathcal L}^m (\mbox{\rm proj}_V E) > 0$ for $\gamma_{n,m}$-almost all $V \in G(n,m)$.
\end{theo}

\begin{theo}\label{marthm}
Let $E \subset \mathbb{R}^2$ be a Borel or analytic set. Then

(i) $\dh \mbox{\rm proj}_\theta E \leq \min\{\dh E, 1\}$ with equality for almost all $\theta \in [0,\pi)$,
\smallskip

(ii) if $\dh E >1$ then ${\mathcal L} (\mbox{\rm proj}_\theta E) > 0$ for almost all $\theta \in [0,\pi)$.
\end{theo}

The  natural higher dimensional analogue of Marstrand's theorem was first presented by Mattila \cite{Mat} in 1975.

\begin{theo}\label{higherdim}
Let $E \subset \mathbb{R}^n$ be a Borel or analytic set. Then

(i) $\dh\mbox{\rm proj}_V E \leq \min\{\dh E, m\}$  with equality for $\gamma_{n,m}$-almost all $V \in G(n,m)$,
\smallskip

(ii) if $\dh E >m$ then ${\mathcal L}^m (\mbox{\rm proj}_V E) > 0$ for $\gamma_{n,m}$-almost all $V \in G(n,m)$.
\end{theo}

Since orthogonal projection is a Lipschitz map that does not increase distances between points, the inequalities in (i) of Theorems \ref{marthm} and \ref{higherdim}   follow easily from the definition of Hausdorff measure and dimension, but showing that equality holds for almost all $\theta$ or $V$ is more involved.
In his original paper, Marstrand  \cite{Mar} used intricate estimates involving plane geometry and measure theory, but in 1968  Kaufman \cite{Kau} gave a new proof of Theorem \ref{marthm}(i) using potential theory and of Theorem \ref{marthm}(ii)  using Fourier transforms before
Mattila \cite{Mat} used this approach in higher dimensions to obtain Theorem 1.2.

Briefly, the potential-theoretic method depends on the characterisation of Hausdorff dimension in terms of energy integrals. We write $\mathcal{M}(E)$ for the positive finite Borel measures supported on $E\subset \mathbb{R}^n$. For $s>0$ we write:
\begin{equation}\label{endef}
I^s(\mu) := \int \!\! \int \frac{d\mu(x)d\mu(y)}{|x-y|^s}
\end{equation}
for the $s$-{\it energy} of the measure $\mu\in \mathcal{M}(E)$. Then
\begin{equation*}\label{endefhau}
\dh E = \sup\Big\{s :  \mbox{ there exists $\mu \in \mathcal{M}(E)$ such that $I^s(\mu) < \infty$}\Big\}.
\end{equation*}
Theorem \ref{marthm}(i) may then be proved by noting that for all $0<s< \dh E$ there is a measure $\mu\in \mathcal{M}(E)$ such that $I^s(\mu) <\infty$,   and verifying that  $\int I^s(\mu_\theta) d\theta \leq c_sI^s(\mu)$, where $\mu_\theta$ is the projection of the measure $\mu$ onto the line $L_\theta$ in direction $\theta$, given by $\mu_\theta(F) = \mu( \mbox{\rm proj}^{-1}_\theta F)$ for $F\subset L_\theta$. The higher dimensional case, Theorem \ref{higherdim}(i), may be proved in a similar way.

Theorem \ref{marthm}(ii) may be obtained using Fourier transforms. Writing $\widehat{\mu}(z):= \int e^{-2\pi i x.z} d\mu(x) \  (z\in \mathbb{R}^n)$ for the Fourier transform of $\mu$, applying Parseval's theorem and the convolution formulae to the energy expression in \eqref{endef} gives
\begin{equation}\label{ftend}
I^s(\mu) = c_{n,s}\int_{\mathbb{R}^n} \frac{|\widehat{\mu}(z)|^2} {|z|^{n-s}}dz \qquad (0<s<n),
\end{equation}
where   $c_{n,s}$ depends only on $n$ and $s$, see \cite[Theorem 3.10]{Mat2}.
Then if $E\subset  \mathbb{R}^2$ and $\dh E >1$  we may find $\mu\in \mathcal{M}(E)$ and $1<s<\dh E$ such that $I^s(\mu) < \infty$. Writing \eqref{ftend} in radial coordinates this implies that $\int \!\! \int | \widehat{\mu_\theta}(t)|^2 dt\, d\theta <\infty$, so that for almost all $\theta$, the projected measure $\mu_\theta$ on $L_\theta$ is absolutely continuous with respect to $\mathcal{L}$ and is a square integrable function, so in particular $E$ has support of positive $\mathcal{L}$-measure.

Energy and Fourier methods and their generalisations and extensions have been used widely across fractal geometry and in particular in work on projections. Further examples will be encountered in this survey, but for thorough treatments see, for example, \cite{Fal, FFJ, Mat, Mat1,Mat4,Mat5,Mat2}

Marstrand's theorem is the prototype for a great deal of work in fractal geometry. If, in some setting, a dimensional property of parameterised images of a set is true for almost all parameters, perhaps for almost all $\theta$ or $V$, we say that a `Marstrand-type theorem holds'. We will see many instances of this in the sections that follow.

For (i), the inequality $\dh\,\mbox{\rm proj}_\theta E\leq \min\{\dh E,1\}$ (and similarly $\dh\,\mbox{\rm proj}_V E\leq \min\{\dh E,m\}$) is said to follow easily since “orthogonal projection is a Lipschitz map that does not increase distances between points”. To get the a.e. equality is “more involved”, and Kaufman’s/Mattila’s approach uses potential theory via energy integrals.

Potential-theoretic step: use the characterisation
\[
\dh E = \sup\Big\{s : \exists\,\mu\in\mathcal M(E)\text{ with }I^s(\mu)<\infty\Big\},\qquad I^s(\mu):=\iint \frac{d\mu(x)\,d\mu(y)}{|x-y|^s}.
\]
Then to prove Theorem~\ref{marthm}(i), “for all $0<s<\dh E$ there is a measure $\mu\in\mathcal M(E)$ such that $I^s(\mu)<\infty$”, and one “verif[ies] that $\int I^s(\mu_\theta)\,d\theta\le c_s I^s(\mu)$”, where $\mu_\theta$ is the projected measure defined by $\mu_\theta(F)=\mu(\mbox{\rm proj}_\theta^{-1}F)$. This yields the a.e. dimension conclusion; “The higher dimensional case, Theorem~\ref{higherdim}(i), may be proved in a similar way.”

For (ii), the sketch uses Fourier transforms: with $\widehat\mu(z)=\int e^{-2\pi i x\cdot z}\,d\mu(x)$, “applying Parseval’s theorem and the convolution formulae” gives
\[
I^s(\mu)=c_{n,s}\int_{\mathbb R^n}\frac{|\widehat\mu(z)|^2}{|z|^{n-s}}\,dz\quad (0<s<n).
\]
If $E\subset\mathbb R^2$ and $\dh E>1$, pick $\mu\in\mathcal M(E)$ and $1<s<\dh E$ with $I^s(\mu)<\infty$. “Writing \eqref{ftend} in radial coordinates this implies that $\iint |\widehat{\mu_\theta}(t)|^2\,dt\,d\theta<\infty$,” hence for a.e. $\theta$ the projected measure $\mu_\theta$ is absolutely continuous w.r.t. $\mathcal L$ and is square-integrable, so “in particular $E$ has support of positive $\mathcal L$-measure” (i.e. ${\mathcal L}(\mbox{\rm proj}_\theta E)>0$ for a.e. $\theta$).

For (i), the inequality $\dh\,\mbox{\rm proj}_\theta E\leq \min\{\dh E,1\}$ (and similarly $\dh\,\mbox{\rm proj}_V E\leq \min\{\dh E,m\}$) is said to follow easily since “orthogonal projection is a Lipschitz map that does not increase distances between points”. To get the a.e. equality is “more involved”, and Kaufman’s/Mattila’s approach uses potential theory via energy integrals.

Potential-theoretic step: use the characterisation
\[
\dh E = \sup\Big\{s : \exists\,\mu\in\mathcal M(E)\text{ with }I^s(\mu)<\infty\Big\},\qquad I^s(\mu):=\iint \frac{d\mu(x)\,d\mu(y)}{|x-y|^s}.
\]
Then, in establishing the main theorem, “for all $0<s<\dh E$ there is a measure $\mu\in\mathcal M(E)$ such that $I^s(\mu)<\infty$”, and one “verif[ies] that $\int I^s(\mu_\theta)\,d\theta\le c_s I^s(\mu)$”, where $\mu_\theta$ is the projected measure defined by $\mu_\theta(F)=\mu(\mbox{\rm proj}_\theta^{-1}F)$. This yields the a.e. dimension conclusion.

We next use the following theorem.
\begin{theo}\label{higherdim}
Let $E \subset \mathbb{R}^n$ be a Borel or analytic set. Then

(i) $\dh\mbox{\rm proj}_V E \leq \min\{\dh E, m\}$  with equality for $\gamma_{n,m}$-almost all $V \in G(n,m)$,
\smallskip

(ii) if $\dh E >m$ then ${\mathcal L}^m (\mbox{\rm proj}_V E) > 0$ for $\gamma_{n,m}$-almost all $V \in G(n,m)$.
\end{theo}

For (ii), the sketch uses Fourier transforms: with $\widehat\mu(z)=\int e^{-2\pi i x\cdot z}\,d\mu(x)$, “applying Parseval’s theorem and the convolution formulae” gives
\begin{equation}\label{ftend}
I^s(\mu) = c_{n,s}\int_{\mathbb{R}^n} \frac{|\widehat{\mu}(z)|^2} {|z|^{n-s}}dz \qquad (0<s<n),
\end{equation}
If $E\subset\mathbb R^2$ and $\dh E>1$, pick $\mu\in\mathcal M(E)$ and $1<s<\dh E$ with $I^s(\mu)<\infty$. “Writing the equation above in radial coordinates this implies that $\iint |\widehat{\mu_\theta}(t)|^2\,dt\,d\theta<\infty$,” hence for a.e. $\theta$ the projected measure $\mu_\theta$ is absolutely continuous w.r.t. $\mathcal L$ and is square-integrable, so “in particular $E$ has support of positive $\mathcal L$-measure” (i.e. ${\mathcal L}(\mbox{\rm proj}_\theta E)>0$ for a.e. $\theta$).

\label{marthm}
Let $E \subset \mathbb{R}^2$ be a Borel or analytic set. Then

(i) $\dh \mbox{\rm proj}_\theta E \leq \min\{\dh E, 1\}$ with equality for almost all $\theta \in [0,\pi)$,
\smallskip

(ii) if $\dh E >1$ then ${\mathcal L} (\mbox{\rm proj}_\theta E) > 0$ for almost all $\theta \in [0,\pi)$.,

Question: Let $E\subset \mathbb{R}^2$ be a Borel or analytic set. For each angle $\theta\in[0,\pi)$, let $L_\theta$ be the line making angle $\theta$ with the $x$-axis, and let $\operatorname{proj}_\theta:\mathbb{R}^2\to L_\theta$ denote orthogonal projection onto $L_\theta$. Write $\dim_H$ for Hausdorff dimension and $\mathcal L$ for Lebesgue measure on the line $L_\theta$ (identified with $\mathbb R$). Which statement holds for every such set $E$?

For every $\theta\in[0,\pi)$, $\dim_H(\operatorname{proj}_\theta E)\le \min\{\dim_H E,1\}$, and equality holds for Lebesgue-almost every $\theta\in[0,\pi)$. Moreover, if $\dim_H E>1$, then $\mathcal L(\operatorname{proj}_\theta E)>0$ for Lebesgue-almost every $\theta\in[0,\pi)$.

Choices:
B. For every $\theta\in[0,\pi)$, $\dim_H(\operatorname{proj}_\theta E)\le \min\{\dim_H E,1\}$, and equality holds for every $\theta\in[0,\pi)$. Moreover, if $\dim_H E\ge 1$, then $\mathcal L(\operatorname{proj}_\theta E)>0$ for Lebesgue-almost every $\theta\in[0,\pi)$.
C. For Lebesgue-almost every $\theta\in[0,\pi)$, $\dim_H(\operatorname{proj}_\theta E)= \min\{\dim_H E,1\}$.
D. For Lebesgue-almost every $\theta\in[0,\pi)$, $\dim_H(\operatorname{proj}_\theta E)\le \min\{\dim_H E,1\}$, and equality holds for every $\theta\in[0,\pi)$. Moreover, if $\dim_H E>1$, then there exists a set of angles of positive Lebesgue measure on which $\mathcal L(\operatorname{proj}_\theta E)>0$.
E. For every $\theta\in[0,\pi)$, $\dim_H(\operatorname{proj}_\theta E)\le \min\{\dim_H E,1\}$, and equality holds for Lebesgue-almost every $\theta\in[0,\pi)$. Moreover, if $\dim_H E>1$, then $\mathcal L(\operatorname{proj}_\theta E)>0$ for every $\theta\in[0,\pi)$.

strict threshold $\dim_H E>1$ replaced by $\dim_H E\ge 1$ and a.e. equality upgraded to all directions

dropped the pointwise Lipschitz upper bound and the positive-Lebesgue-measure conclusion

quantifier placement on exceptional directions altered: upper bound only a.e., equality strengthened to every direction, positivity weakened from a.e. to merely positive-measure set of angles

almost-everywhere positivity conclusion strengthened to every direction

The stem does not explicitly or implicitly reveal choice A; the solver must distinguish between subtle quantifier and threshold variations across the options.

The item is largely a recognition/recollection of Marstrand-type projection statements, but it is not a pure tautological restatement because the choices differ in meaningful ways such as 'for every' versus 'for almost every' and '>1' versus '>=1'.

Some reasoning is needed to track the sharp form of the theorem and reject overstatements, but the task is still mainly theorem recall rather than substantial generative mathematical reasoning. Also, the presence of a weaker true option reduces pressure.

B, D, and E are plausible distractors built from common quantifier mistakes, but C is also true, making the distractor set flawed for a single-answer MCQ and weakening overall option quality.

A reasonably well-targeted theorem-discrimination item with good quantifier-based traps, but it is compromised as a single-answer MCQ because choice C is also true, creating ambiguity and reducing its effectiveness.


Let $p$ be a prime, and let $\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\mathbb{Z}$ as
\[ I_i=[N_i+1,N_i+H_i]\cap \mathbb{Z}.\]  Fix a basis $\{\omega_{1},\ldots,\omega_{n}\}$ of $\mathbb{F}_{p^{n}}$ over $\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying
\[
1 \leq H_i \leq p \qquad (1 \leq i \leq n),
\]
we define the box
\[
B=\left\{ \sum_{i=1}^{n} x_i\omega_i :
N_i+1 \leq x_i \leq N_i+H_i,\; 1 \leq i \leq n \right\}
\subseteq \mathbb{F}_{p^{n}}.
\]
and the volume of the box as
\[|B|=H_1\cdot H_2 \cdots H_n. \]
Let $\chi$ be a multiplicative character on $\mathbb{F}_{p^n}$.

\medskip

We begin with some historical background. In the case $n=1$, Burgess's bound \cite{B1} remains the strongest result to date. It asserts that for every $\varepsilon>0$ there exists $\delta>0$ such that
\[
\left|\sum_{x=N+1}^{N+H}\chi(x)\right|\ll_{\varepsilon} p^{-\delta}H
\]
whenever $H\geq p^{1/4+\varepsilon}$. In subsequent work \cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \cite{K} later extended these ideas to general finite fields $\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

For arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \cite{DL}. They studied character sums over $\mathbb{F}_{p^n}$ of the form
\[
\sum_{x_1\in I_1,\ldots,x_d\in I_n} \chi\!\left(x_1\omega_1+\cdots+x_n\omega_n\right),
\]
where $\{\omega_1,\ldots,\omega_n\}$ is a fixed basis of $\mathbb{F}_{p^d}$ over $\mathbb{F}_p$ and $I_i$ are intervals in $\mathbb{Z}$. They proved that
\[
\sum_{\mathbf{x}\in B}\chi(\mathbf{x})
=O\!\left(H_1\cdots H_n\,p^{-\delta(\varepsilon)}\right),
\]
provided that $H_i=p^{\rho_n+\varepsilon}$ for all $i$, where
\[
\rho_d=\frac12-\frac{1}{2(d+1)}.
\]

This result was later improved by M. C. Chang \cite{C1}, who showed that
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{d,\varepsilon} p^{-\varepsilon^2/4}|B|,
\]
whenever $|H_i|>p^{2/5+\varepsilon}$ for all $1\leq i\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\geq5$. Chang \cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\varepsilon}$.

Subsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.

Let $p$ be a prime, and let $\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\mathbb{Z}$ as
\[ I_i=[N_i+1,N_i+H_i]\cap \mathbb{Z}.\] Fix a basis $\{\omega_{1},\ldots,\omega_{n}\}$ of $\mathbb{F}_{p^{n}}$ over $\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying
\[
1 \leq H_i \leq p \qquad (1 \leq i \leq n),
\]
we define the box
\[
B=\left\{ \sum_{i=1}^{n} x_i\omega_i :
N_i+1 \leq x_i \leq N_i+H_i,\; 1 \leq i \leq n \right\}
\subseteq \mathbb{F}_{p^{n}}.
\]
and the volume of the box as
\[|B|=H_1\cdot H_2 \cdots H_n. \]
Let $\chi$ be a multiplicative character on $\mathbb{F}_{p^n}$.

We begin with some historical background. In the case $n=1$, Burgess's bound \cite{B1} remains the strongest result to date. It asserts that for every $\varepsilon>0$ there exists $\delta>0$ such that
\[
\left|\sum_{x=N+1}^{N+H}\chi(x)\right|\ll_{\varepsilon} p^{-\delta}H
\]
whenever $H\geq p^{1/4+\varepsilon}$. In subsequent work \cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \cite{K} later extended these ideas to general finite fields $\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

For arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \cite{DL}. They studied character sums over $\mathbb{F}_{p^n}$ of the form
\[
\sum_{x_1\in I_1,\ldots,x_d\in I_n} \chi\!\left(x_1\omega_1+\cdots+x_n\omega_n\right),
\]
where $\{\omega_1,\ldots,\omega_n\}$ is a fixed basis of $\mathbb{F}_{p^d}$ over $\mathbb{F}_p$ and $I_i$ are intervals in $\mathbb{Z}$. They proved that
\[
\sum_{\mathbf{x}\in B}\chi(\mathbf{x})
=O\!\left(H_1\cdots H_n\,p^{-\delta(\varepsilon)}\right),
\]
provided that $H_i=p^{\rho_n+\varepsilon}$ for all $i$, where
\[
\rho_d=\frac12-\frac{1}{2(d+1)}.
\]

This result was later improved by M. C. Chang \cite{C1}, who showed that
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{d,\varepsilon} p^{-\varepsilon^2/4}|B|,
\]
whenever $|H_i|>p^{2/5+\varepsilon}$ for all $1\leq i\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\geq5$. Chang \cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\varepsilon}$.

Subsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.

We begin with some historical background. In the case $n=1$, Burgess's bound \cite{B1} remains the strongest result to date. It asserts that for every $\varepsilon>0$ there exists $\delta>0$ such that
\[
\left|\sum_{x=N+1}^{N+H}\chi(x)\right|\ll_{\varepsilon} p^{-\delta}H
\]
whenever $H\geq p^{1/4+\varepsilon}$. In subsequent work \cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \cite{K} later extended these ideas to general finite fields $\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

For arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \cite{DL}. They studied character sums over $\mathbb{F}_{p^n}$ of the form
\[
\sum_{x_1\in I_1,\ldots,x_d\in I_n} \chi\!\left(x_1\omega_1+\cdots+x_n\omega_n\right),
\]
where $\{\omega_1,\ldots,\omega_n\}$ is a fixed basis of $\mathbb{F}_{p^d}$ over $\mathbb{F}_p$ and $I_i$ are intervals in $\mathbb{Z}$. They proved that
\[
\sum_{\mathbf{x}\in B}\chi(\mathbf{x})
=O\!\left(H_1\cdots H_n\,p^{-\delta(\varepsilon)}\right),
\]
provided that $H_i=p^{\rho_n+\varepsilon}$ for all $i$, where
\[
\rho_d=\frac12-\frac{1}{2(d+1)}.
\]

This result was later improved by M. C. Chang \cite{C1}, who showed that
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{d,\varepsilon} p^{-\varepsilon^2/4}|B|,
\]
whenever $|H_i|>p^{2/5+\varepsilon}$ for all $1\leq i\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\geq5$. Chang \cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\varepsilon}$.

Subsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.

While this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.

\begin{theorem}[Gabdullin \cite{GB}]
Let $n\in\{2,3\}$, let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, and suppose
\[
|B|\geq p^{n(1/4+\varepsilon)}, \qquad H_1\leq H_2\leq\cdots\leq H_n.
\]
Then
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{n,\varepsilon} |B| p^{-\varepsilon^2/12}.
\]
\end{theorem}

\begin{theorem}[Main Theorem]\label{MT}
Let $n\geq2$, let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, and suppose
\[
|B|\geq p^{n(1/4+\varepsilon)},\qquad
H_1\leq H_2\leq\cdots\leq H_n\leq \sqrt{p/2}.
\]
Then
\[
\left|\sum_{x\in B}\chi(x)\right|
\ll |B|\, p^{-\varepsilon^2
\frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}}.
\]
\end{theorem}

As a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$, in contrast with our previous work.
\begin{Corollary}
Let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, let $I_i \subset \mathbb{Z}$ be intervals of lengths $H_i$, and let $\{\omega_1,\ldots,\omega_n\}$ be a basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$.
For every $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that
\[
\left|\sum_{x_1\in I_1,\ldots,x_k\in I_k}
\chi(x_1\omega_1+\cdots+x_k\omega_k)\right|
\ll_{n,\varepsilon}
H_1\cdots H_k\,p^{-\delta(\varepsilon)},
\]
provided that
\[
H_i \ge p^{\frac{n}{k}\left(\frac14+\varepsilon\right)}
\quad (1\le i\le k),
\]
where
\[
\delta(\varepsilon)
=
\varepsilon^2
\frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}.
\]
In particular, if $k=n-o(n)$, then the condition becomes
$H_i\geq p^{\frac{n}{n-o(n)}(1/4+\varepsilon)}$, and this exponent tends to $1/4+\varepsilon$ as $n\to\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.
\end{Corollary}

While this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.

\begin{theorem}[Gabdullin \cite{GB}]
Let $n\in\{2,3\}$, let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, and suppose
\[
|B|\geq p^{n(1/4+\varepsilon)}, \qquad H_1\leq H_2\leq\cdots\leq H_n.
\]
Then
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{n,\varepsilon} |B| p^{-\varepsilon^2/12}.
\]
\end{theorem}

In this paper, we generalize Gabdullin's result to arbitrary dimension $n$. Our main theorem establishes nontrivial cancellation for character sums over general boxes in $\mathbb{F}_{p^n}$, assuming only a lower bound on the volume of the box.

\begin{theorem}[Main Theorem]\label{MT}
Let $n\geq2$, let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, and suppose
\[
|B|\geq p^{n(1/4+\varepsilon)},\qquad
H_1\leq H_2\leq\cdots\leq H_n\leq \sqrt{p/2}.
\]
Then
\[
\left|\sum_{x\in B}\chi(x)\right|
\ll |B|\, p^{-\varepsilon^2
\frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}}.
\]
\end{theorem}

\begin{remark}
If the extension degree $n$ is prime, the monotonicity assumption $H_1\leq\cdots\leq H_n\leq \sqrt{p/2}$ may be removed. A sketch of the argument is given in Case 3 of the proof of Theorem \ref{MT}.
\end{remark}

In a previous work \cite{Ch}, the author studied character sums over sublattices of codimension one in $\mathbb{F}_{p^n}$, obtaining nontrivial bounds under significantly weaker length conditions.

\begin{theorem}[Chattopadhyay \cite{Ch}]
Let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^d}$, and let $\omega$ be a generator of $\mathbb{F}_{p^n}^\ast$. Given $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that if $I_0,\ldots,I_{n-2}$ are intervals of length $p^{\rho''_n+\varepsilon}$, where
\[
\rho''_n=\frac{1460-1000n+\sqrt{1000000n^2-1960000n+490000}}{960},
\]
then
\[
\sum_{x_0\in I_0,\ldots,x_{n-2}\in I_{n-2}}
\chi(x_0+x_1\omega+\cdots+x_{n-2}\omega^{n-2})
\ll p^{-\delta(\varepsilon)} |I_0|\cdots|I_{n-2}|.
\]
\end{theorem}

The novelty of this result lied in treating character sums over sublattices of codimension one.

\medskip

As a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$, in contrast with our previous work.
\begin{Corollary}
Let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, let $I_i \subset \mathbb{Z}$ be intervals of lengths $H_i$, and let $\{\omega_1,\ldots,\omega_n\}$ be a basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$.
For every $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that
\[
\left|\sum_{x_1\in I_1,\ldots,x_k\in I_k}
\chi(x_1\omega_1+\cdots+x_k\omega_k)\right|
\ll_{n,\varepsilon}
H_1\cdots H_k\,p^{-\delta(\varepsilon)},
\]
provided that
\[
H_i \ge p^{\frac{n}{k}\left(\frac14+\varepsilon\right)}
\quad (1\le i\le k),
\]
where
\[
\delta(\varepsilon)
=
\varepsilon^2
\frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}.
\]
In particular, if $k=n-o(n)$, then the condition becomes
$H_i\geq p^{\frac{n}{n-o(n)}(1/4+\varepsilon)}$, and this exponent tends to $1/4+\varepsilon$ as $n\to\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.
\end{Corollary}

\medskip

The proof of the Main Theorem closely follows the approach of \cite{GB}, employing techniques from \cite{C1}, \cite{K} and \cite{Kon} to estimate character sums over boxes in finite fields whose side lengths are at most $\sqrt{p/2}$. Our argument begins with the classical additive shifting technique $x\mapsto x+yz$ introduced by Burgess \cite{B2}. Applying H\"older's inequality twice to the resulting sum of shifted characters produces three terms: one is related to the multiplicative energy of the box, while the remaining terms can be bounded using Weil's bound and trivial estimates. Consequently, achieving a nontrivial saving for the character sum reduces to obtaining a bound on the multiplicative energy of a set in $\mathbb{F}_{p^n}$ which are described in Lemma \ref{l1} and Lemma \ref{l2}. More specifically, we employ tools from the geometry of numbers, in particular the concepts of successive minima and Minkowski's second theorem, together with an auxiliary lemma that provides upper and lower bounds for the product of successive minimas which turns out to be an essential ingredient in our energy estimate.\\

In fact, we have also removed the boundedness assumption on the side lengths of the boxes in Theorem~\ref{MT}, as described in \textbf{Remark~1}, when the degree of the extension is prime. The key ingredient in the proof of \textbf{Remark~1} is Theorem~\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\mathbb{F}_{p}$. Under the remaining assumption that the larger side length of the box exceeds $p^{1/2+\varepsilon}$, we obtain a non-trivial saving for the character sum along that direction. Estimating the remaining summations trivially then yields an overall non-trivial saving over the volume of the box. For further details, see \textbf{Case~3} in the proof of the main theorem.

\medskip

{\bf Acknowledgments.} The author expresses his sincere gratitude to Professor Stephan Baier for his valuable conversations. The author thanks the Ramakrishna Mission Vivekananda Educational and Research Institute for an excellent work environment. The research of the author was supported by a CSIR Ph.D. fellowship under file number 09/0934(13170)/2022-EMR-I.

The proof of Theorem~\ref{MT} “closely follows the approach of \cite{GB},” using “techniques from \cite{C1}, \cite{K} and \cite{Kon}.” It begins with “the classical additive shifting technique $x\mapsto x+yz$ introduced by Burgess \cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\sqrt{p/2}$”). Then, “applying H\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\mathbb{F}_{p^n}$,” as given in “Lemma~\ref{l1} and Lemma~\ref{l2}.” For the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.

For the prime-degree refinement mentioned after Theorem~\ref{MT} (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is “Theorem~\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\mathbb{F}_{p}$.” If “the larger side length of the box exceeds $p^{1/2+\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).

The proof of
\begin{theorem}[Main Theorem]\label{MT}
Let $n\geq2$, let $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$, and suppose
\[
|B|\geq p^{n(1/4+\varepsilon)},\qquad
H_1\leq H_2\leq\cdots\leq H_n\leq \sqrt{p/2}.
\]
Then
\[
\left|\sum_{x\in B}\chi(x)\right|
\ll |B|\, p^{-\varepsilon^2
\frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}}.
\]
\end{theorem}
“closely follows the approach of \cite{GB},” using “techniques from \cite{C1}, \cite{K} and \cite{Kon}.” It begins with “the classical additive shifting technique $x\mapsto x+yz$ introduced by Burgess \cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\sqrt{p/2}$”). Then, “applying H\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\mathbb{F}_{p^n}$,” as given in the following lemmas.

\begin{Lemma}\label{l1}
We have
\[
L_{1} \;\ll\; |B|^{2}\log p.
\]
\end{Lemma}

\begin{Lemma}\label{l2}
We have
\[
L_{2} \;\ll\; |B|^{2}(\log p)^{n}.
\]
\end{Lemma}

For the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.

For the prime-degree refinement mentioned after the main theorem (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is the following proposition, which provides square-root cancellation for the corresponding complete character sum over $\mathbb{F}_{p}$.

\begin{Proposition}[Katz]\label{KT}
Let $\chi$ be a non-trivial multiplicative character of $\mathbb{F}_{p^n}$ and let
$g\in\mathbb{F}_{p^n}$ generate the extension
$\mathbb{F}_{p^n}=\mathbb{F}_p(g)$. Then, for any interval
$I\subseteq [1,p]\cap\mathbb{Z}$,
\[
\left|\sum_{t\in I}\chi(g+t)\right|
\le c(n)\sqrt{p}\log p .
\]
\end{Proposition}

If “the larger side length of the box exceeds $p^{1/2+\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).

[Konyagin \cite{Kon}]
Let $\varepsilon>0$ and suppose $H_i>p^{1/4+\varepsilon}$ for all $1\leq i\leq n$. Then
\[
\left|\sum_{\mathbf{x}\in B}\chi(\mathbf{x})\right|
\ll_{n,\varepsilon} p^{-\varepsilon^2/2}|B|.
\],

Question:
Let p be a prime, let \(\mathbb{F}_{p^n}\) be the finite field with \(p^n\) elements, and fix a basis \(\{\omega_1,\ldots,\omega_n\}\) of \(\mathbb{F}_{p^n}\) over \(\mathbb{F}_p\). For integers \(N_i\) and \(H_i\), define intervals \(I_i=[N_i+1,N_i+H_i]\cap\mathbb{Z}\) and the box
\[
B=\left\{\sum_{i=1}^n x_i\omega_i: N_i+1\le x_i\le N_i+H_i\text{ for }1\le i\le n\right\}\subseteq \mathbb{F}_{p^n},
\]
with volume \(|B|=H_1\cdots H_n\). Let \(\chi\) be a nontrivial multiplicative character on \(\mathbb{F}_{p^n}\). If \(\varepsilon>0\) and \(H_i>p^{1/4+\varepsilon}\) for every \(1\le i\le n\), which quantitative estimate holds for the character sum over \(B\)?

\[\left|\sum_{x\in B}\chi(x)\right|\ll_{n,\varepsilon} p^{-\varepsilon^2/2}\,|B|.\]

Choices:
B. \[\left|\sum_{x\in B}\chi(x)\right|\ll_{n,\varepsilon} p^{-\varepsilon^2/2}\,|B|,\qquad \text{provided }H_i\ge p^{1/4+\varepsilon}\text{ for every }1\le i\le n.\]
C. \[\left|\sum_{x\in B}\chi(x)\right|=o_{n,\varepsilon}(|B|)\quad\text{as }p\to\infty.\]
D. \[\left|\sum_{x\in B}\chi(x)\right|\ll_{\varepsilon} p^{-\varepsilon^2/2}\,|B|.\]
E. \[\left|\sum_{x\in B}\chi(x)\right|\ll_{n,\varepsilon} p^{-\varepsilon^2}\,|B|.\]

explicit power saving factor p^{-\varepsilon^2/2}

The stem gives the setup and hypotheses but does not disclose the exact conclusion. The correct option is not signaled by wording in the question itself.

This is close to theorem recall: the task is essentially to identify the precise stated conclusion under the given hypotheses. However, it is not a verbatim restatement because the student must distinguish among nearby alternatives.

Some reasoning is needed to compare exact, weaker, and stronger bounds and note subtle quantifier issues, but the item mainly tests recognition of the known result rather than genuinely generating an argument.

The distractors are mathematically plausible and well targeted: one is weaker but true, one adds unnecessary side conditions, one alters dependence of the saving exponent, and one overstates the exponent. These reflect realistic failure modes.

A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than toward deep generative reasoning.


\label{thm:maintheorem}(cf. Theorem \ref{thm:cdbound})
Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\operatorname{cd}(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$

\begin{lemma}\label{lemma:p_zd_on_H_I^n}
    Let $I$, $A$, $x$, $I_1$ and $I_2$ be as in Lemma \ref{lemma:radical(I+p)}, $M$ be an $A$-module and $n>\max(\cd(M,I_1),\cd(M,I_2))$. Then $H_I^n(M)$ is $x$-power torsion, that is, every element of $H^n_I(M)$ is annihilated by some power of $x$.
\end{lemma}

\begin{maintheorem}\label{thm:maintheorem}(cf. Theorem \ref{thm:cdbound})
Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\cd(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$
\end{maintheorem}

\begin{maintheorem}\label{thm:indcrit}(cf. Theorem \ref{thm:induction})
    Let $(A,\mathfrak{m})$ be an unramified regular local ring of mixed characteristic $(0,p)$ and $I$ be an ideal of $A$ of big height $c$. Suppose $p$ is a nonzerodivisor on $A/I.$

    Let $n>c$ be an integer. Assume that for all integers $s$, with $1\leq s\leq c-1,$ and for all $q\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\in \Spec(A)$ such that $I\subseteq P$ and $\dim(A/P)\geq s+1.$ Then $H^q_I(A)=0$ for all $q\geq n.$
\end{maintheorem}

Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem.

Let $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \textit{cohomological dimension} of $I$ in $A$ is $$\operatorname{cd}(A,I):=\max\{n\:|\:H^n_I(A)\neq 0\}.$$ Alternatively, $\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \cite{Gr}, $\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.

The computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\operatorname{cd}(A,I)\leq\operatorname{ara}(I),$$ where $\operatorname{ara}(I)$, the $\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \cite{BS,Bar,JPSW,BMMP}.

A complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \cite{Ha1}, which states that for a complete local domain $(A,\mathfrak{m})$, we have $\operatorname{cd}(A,I)\leq d-1$ if and only if $I$ is not $\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\mathfrak{m})$ with separably closed residue field, we have $\operatorname{cd}(A,I)\leq d-2$ if and only if $\dim(A/I)\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \cite{PS} proved that if $(A,\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\operatorname{cd}(A,I)=\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \cite{Ogus}, Hartshorne and Speiser \cite{HarSpe}, Lyubeznik \cite{Lyu3}, Mustață and Popa \cite{MuPo}, and Reichelt, Saito and Walther \cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.

In equal characteristic, Faltings \cite{Faltings} established a general bound on cohomological dimension in terms of the \textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.

\begin{theorem}\label{thm:cdbound}
    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\cd(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$
\end{theorem}

In equal characteristic, Faltings \cite{Faltings} established a general bound on cohomological dimension in terms of the \textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.

We also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \ref{thm:induction} for a more general statement than the one stated below.

\begin{theorem}\label{thm:cdbound}
 Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\cd(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$
\end{theorem}
\begin{proof} We may assume that $A$ is complete and $I$ is radical as local cohomology and big height remain unchanged. If $c=1$, $I$ is principal and $\cd(A,I)\leq 1$. Thus we assume $c>1.$ We also assume $c<d$ as the claimed bound is immediate if $c=d.$

\begin{corollary}[cf. {\cite[Lemma]{Lyu2}}, {\cite[Lemma 3.10]{HNBPW}}]\label{cor:Lyu}
 Let $A$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Let $I_1,\dots,I_N$ be ideals of $A$ of big height at most $c\geq 1$. Then $$\cd(A,I_1+\dots+I_N)\leq d-\floor*{\frac{d-1}{c}}+N-1.$$
\end{corollary}
\begin{proof}
 We proceed by induction on $N$. The case $N=1$ follows from \cite[Satz 1]{Faltings} in equal characteristic and Theorem \ref{thm:cdbound} in unramified mixed characteristic. For $N>1,$ let $I'=I_1+\dots+I_{N-1}$, $I=I'+I_N$ and $J=I_1\cap I_t+\dots+I_{N-1}\cap I_N$. Note that each $I_j\cap I_t$ has big height at most $c$ by Remark \ref{rmk:bight}. As $\sqrt{J}=\sqrt{I'\cap I_N}$, the Mayer-Vietoris sequence for local cohomology yields $$\cdots\to H_J^{i-1}(A)\to H_I^{i}(A)\to H_{I'}^{i}(A)\oplus H_{I_N}^{i}(A)\to \cdots$$
 For $i>d-\floor*{\frac{d-1}{c}}+N-1$, $H_J^{i-1}(A)=H_{I'}^{i}(A)=H_{I_N}^{i}(A)=0$ by the induction hypothesis. Hence, $H_I^i(A)=0$ for $i>d-\floor*{\frac{d-1}{c}}+N-1$.
\end{proof}

\begin{theorem}[cf. {\cite[Theorem 5.2]{HunLyu}}]\label{thm:example}
 Let $(A,\mathfrak{m})$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Let $c<d$ be a positive integer and set $N=\floor*{\frac{d-1}{c}}$. Let $I_1,\dots,I_N$ be ideals of $A$ of big height at most $c$ such that $\Spec(A/(I_1+\dots+I_N))-\{\mathfrak{m}\}$ is disconnected. Then $\cd(A,I_1\cap\dots\cap I_N)=d-N.$
\end{theorem}
\begin{proof}
 If $(B,\mathfrak{n})$ is a faithfully flat $A$-algebra with zero-dimensional fiber, then $\bight(I_jB)=\bight(I)$ (Lemma \ref{lemma:fflat}) and $\Spec(B/(I_1B+\cdots+I_NB)-\{\mathfrak{n}\}$ is disconnected. Hence, we may assume that $A$ has separably closed residue field.

\begin{example}[{\cite[Example 5.4]{HunLyu}}]
 Let $(A,\mathfrak{m})$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Suppose $P_1,\dots,P_5$ are prime ideals of $A$ of height $c$ such that $\floor*{\frac{d-1}{c}}=2.$ Further, suppose that $P_i+P_j+P_k$ is $\mathfrak{m}$-primary iff $$\{i,j,k\}\in\Lambda=\{(1,3,4),(1,3,5),(1,2,5),(2,3,4),(2,4,5)\}.$$ Let $I=P_1\cap\cdots\cap P_5.$
 For instance, we may consider $A=V[[x_2,\dots,x_9]]$, where $V$ is a discrete valuation ring with uniformizer $p$ and separably closed residue field, and $$P_1=(p,x_2,x_3,x_4), \;P_2=(x_3,x_4,x_5,x_6),\; P_3=(x_5,x_6,x_7,x_8),$$ $$P_4=(p,x_2,x_8,x_9) \text{ and } P_5=(x_6,x_7,x_8,x_9).$$
 Note that $\Lambda$ has the property that each $i$, $1\leq i\leq5$ appears no more than thrice and each pair $(i,j)$ appears no more than twice. This implies that we cannot split $\{1,\dots,5\}$ into three distinct subsets $S_0$, $S_1$ and $S_2$ such that $P_i+P_j+P_k$ is $\mathfrak{m}$-primary for all $i\in S_0$, $j\in S_1$, $k\in S_2$. Hence, we are unable to apply the result of Example \ref{example:sharpness}.

\begin{theorem}[{\cite[Theorem 3.8]{HunLyu}}]\label{thm:HunLyu_irred}
 Let $A$ be a $d$-dimensional regular local ring of equal characteristic and $I$ be a formally geometrically irreducible ideal of $A$ of big height $c$. Suppose $0<c<d.$ Then $$\cd(A,I)\leq d-1-\floor*{\frac{d-2}{c}}.$$
\end{theorem}

\begin{question}\label{question:irredbound}
 Let $A$ be a $d$-dimensional regular local ring of unramified mixed characteristic and $I$ be a formally geometrically irreducible ideal of $A$ of big height $c$. Suppose $0<c<d$. Is $$\cd(A,I)\leq d-1-\floor*{\frac{d-2}{c}}?$$
\end{question}

\begin{example}\label{example:sharpness}
    Given an unramified regular local ring $(A,\mathfrak{m})$ of dimension $d>0$ and a positive integer $c\leq d$, there exists an ideal $I$ of $A$ such that $\cd(A,I)=d-\floor*{\frac{d-1}{c}}.$ We construct this ideal as in \cite{Lyu2}.

    Let $N=\floor*{\frac{d-1}{c}}.$ Let $I_0,\dots,I_N$ be ideals of pure height $c$ in $A$ such that $I_0+\dots+I_N$ is $\mathfrak{m}$-primary and let $I=I_0\cap\dots\cap I_N$. Then $\cd(A,I)=d-N.$ The proof of this statement is identical to \cite{Lyu2}.

    To resolve any conflict with the previous remark, we show that in this example, if $p\in I$, then $\floor*{\frac{d-2}{c-1}}= \floor*{\frac{d-1}{c}}.$ Indeed, if $p\in I,$ we have $p\in I_j$ for all $j\in\{0,\dots,N\}$ and by Theorem \ref{thm:Serre}, $$d=\height(I_0+\dots+I_N)\leq 1+(N+1)(c-1)=Nc+c-N.$$ Let $d-1=Nc+k$, where $0\leq k<c$. Then the above equation yields $N+k+1\leq c$, or $N+k-1\leq c-2$. Observe that $$d-2=N(c-1)+N+k-1,$$ and hence, $\floor*{\frac{d-2}{c-1}}=N=\floor*{\frac{d-1}{c}}$.
\end{example}

\begin{theorem}\label{thm:cdbound}
    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\cd(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$
\end{theorem}

\begin{theorem}\label{thm:induction}
    Let $(A,\mathfrak{m},k)$ be an unramified regular local ring of mixed characteristic $(0,p)$, and let $I$ be an ideal of $A$. Suppose $\height(I)<\height(I:(I:p))$ (for example, this holds if $p$ is a nonzerodivisor on $A/I)$.\footnote{We use the convention that $\height(A)=\infty.$}

    Set $c=\bight(I)$. Let $n>c$ be an integer. Assume that for all integers $s$, with $1\leq s\leq c-1,$ and for all $q\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\in \Spec(A)$ such that $I\subseteq P$ and $\dim(A/P)\geq s+1.$ Then $H^q_I(A)=0$ for all $q\geq n.$
\end{theorem}

We also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \ref{thm:induction} for a more general statement than the one stated below.

\begin{maintheorem}\label{thm:indcrit}(cf. Theorem \ref{thm:induction})
    Let $(A,\mathfrak{m})$ be an unramified regular local ring of mixed characteristic $(0,p)$ and $I$ be an ideal of $A$ of big height $c$. Suppose $p$ is a nonzerodivisor on $A/I.$

    Let $n>c$ be an integer. Assume that for all integers $s$, with $1\leq s\leq c-1,$ and for all $q\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\in \operatorname{Spec}(A)$ such that $I\subseteq P$ and $\dim(A/P)\geq s+1.$ Then $H^q_I(A)=0$ for all $q\geq n.$
\end{maintheorem}

We now outline the strategy for proving our results.

\begin{enumerate}
    \item In equal characteristic, Faltings proved a criterion \cite[Satz 1]{Faltings} of the nature of Theorem \ref{thm:indcrit}, and used it as an inductive tool to establish an upper bound on $\operatorname{cd}(A,I).$ We generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$.\footnote{This assumption can be further relaxed: see Theorem \ref{thm:induction} for details.}

    In the proof, we reduce the problem to the case of a local cohomology module supported only at the maximal ideal. We handle this remaining case by introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal. This allows us to exploit precise dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting.
    We then leverage a result of Zhou on the injective dimension of local cohomology modules in unramified mixed characteristic \cite[Theorem 5.1]{Zhou}, together with the finiteness of Bass numbers for such modules established in \cite{Lyu4, NB}.
    \item We split our proof of Theorem \ref{thm:maintheorem} into three cases. If $p$ is a nonzerodivisor on $A/I$, we can use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound. If $p\in I,$ we reduce our analysis to $A/p$, a regular local ring of equal characteristic $p$, and employ the already known bound in equal characteristic.
    \item If $p$ is a zerodivisor on $I$, we build on the analysis of the preceding two cases to once again reduce to equal characteristic $p$. The guiding observation is that $H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound (Lemma \ref{lemma:p_zd_on_H_I^n}).

    At this stage, the argument must be refined to control the behavior of the big height of the ideal $I(A/p).$ A systemic analysis reveals that all cases can be resolved except when $\operatorname{bight}(I)$ divides $\dim(A)-1$. In this remaining case, a key input is the structural description of the critical local cohomology module $H^q_J(A/p)$ \cite[Theorem 4.1]{Lyu1}, which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.

\end{enumerate}

The paper is organized as follows. In Section \ref{section: InductionTheorem}, we collect the principal properties and a number of subtle features of the big height of an ideal. We then prove the inductive criteria for the vanishing of local cohomology in the unramified mixed characteristic setting (Theorem \ref{thm:induction}). In Section \ref{section:cdbound}, we use Theorem \ref{thm:induction} to prove the upper bound on cohomological dimension (Theorem \ref{thm:cdbound}). We also address the sharpness of our bound (Example \ref{example:sharpness}) and illustrate how the bound can be used to compute the exact cohomological dimension of suitably chosen ideals. In Section \ref{section:questions}, we collect natural questions suggested by our results, and comment on current progress and related directions.

To prove Theorem~\ref{thm:maintheorem}, the authors first “generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$,” i.e. they prove an inductive vanishing criterion of the form Theorem~\ref{thm:indcrit}. In that proof, they “reduce the problem to the case of a local cohomology module supported only at the maximal ideal,” then handle it by “introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal,” using “dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting,” and finally “leverage a result of Zhou on the injective dimension of local cohomology modules… together with the finiteness of Bass numbers.”

They then “split [the] proof of Theorem~\ref{thm:maintheorem} into three cases:”
(1) If $p$ is a nonzerodivisor on $A/I$, they “use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound.”
(2) If “$p\in I$,” they “reduce [the] analysis to $A/p$… of equal characteristic $p$, and employ the already known bound in equal characteristic.”
(3) If $p$ is a zerodivisor on $A/I$, they “once again reduce to equal characteristic $p$,” using that “$H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound,” and then refine the argument to control “the behavior of the big height of the ideal $I(A/p)$”; after a “systemic analysis” they reduce remaining cases (notably when $\operatorname{bight}(I)$ divides $\dim(A)-1$) using “the structural description of the critical local cohomology module $H^q_J(A/p)$… which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.”

\label{thm:maintheorem}(To prove the main theorem, we use the following result. \begin{theorem}\label{thm:cdbound}
    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\cd(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$
\end{theorem})
Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\operatorname{cd}(A,I)\leq d-\floor*{\frac{d-1}{c}}.$$,

Question: Let \(A\) be a \(d\)-dimensional unramified regular local ring of mixed characteristic \((0,p)\), and let \(I\subset A\) be a nonzero proper ideal. Write \(c=\operatorname{bight}(I)\), where the big height is the maximum of the heights of the minimal primes of \(I\). Also, let \(\operatorname{cd}(A,I)=\max\{n\mid H_I^n(A)\neq 0\}\) denote the cohomological dimension of \(I\) in \(A\). Which statement holds for every such pair \((A,I)\)?

\(\operatorname{cd}(A,I)\le d-\left\lfloor \frac{d-1}{c}\right\rfloor.\)

Choices:
B. \(\operatorname{cd}(A,I)\le d-\left\lfloor \frac{d}{c}\right\rfloor.\)
C. \(\operatorname{cd}(A,I)\le d-c+1.\)
D. If \(c<d\), then \(\operatorname{cd}(A,I)\le d-1-\left\lfloor \frac{d-2}{c}\right\rfloor.\)
E. \(\operatorname{cd}(A,I)= d-\left\lfloor \frac{d-1}{c}\right\rfloor.\)

replace Faltings-type floor bound by a coarser linear bound implied by it

imports the formally geometrically irreducible-type stronger bound without its extra hypothesis

upper bound promoted to equality for all ideals

The stem does not reveal the target inequality; it only sets up notation and assumptions. There is no explicit statement of the correct bound or a giveaway phrase pointing uniquely to choice A.

The item is largely a theorem-recall question: it asks which estimate holds under exactly the stated hypotheses. The nearby variants create some choice pressure, but the task is still close to selecting the memorized theorem statement rather than deriving a new conclusion.

Some reasoning is needed to distinguish among very similar formulas and to reject weaker/incorrect variants, but the main burden is recognition of the precise bound. It does not strongly force generative mathematical reasoning from first principles.

The distractors are mathematically plausible and target realistic confusions: changing the floor numerator, replacing big height by height, offering a weaker true bound, and using a bound from a related setting. They are distinct and nontrivial.

A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recall rather than deep generative reasoning.