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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


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}


\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


\label{thm:simply}
  Let $S$ be a surface of infinite-type with non-planar ends. Then $\Gamma_k(S)$ is connected and simply connected for each $k\in \mathbb{N}$.

\label{aut}
In this section we prove Theorem \ref{thm:geometric} and henceforth assume that $S$ has no planar ends and boundary. We state the results proved in \cite{irmak2004} without proof whenever

\begin{definition}
  Let $S$ be an infinite-type surface with non-planar ends, $k\in \N$, and let $\Gamma^0_k(S)$ be the collection of cut systems on $S$ with exactly $k$ curves. Let $\Gamma^1_k(S)$ be the graph obtained by attaching a 1-cell between two cut systems in $\Gamma^0_k(S)$ whenever they are related by an elementary move. The generalized Hatcher--Thurston complex, $\Gamma_k(S)$, is obtained by attaching 2-cells to every cycle of type $1,\ 2,$ or $3$.
\end{definition}
Note that $\Gamma^1_1(S)$ is just the Schmutz graph $ \mathcal{G}(S)$. Since the Schmutz graph is connected and $\text{diam}( \mathcal{G}(S))\leq 4$ we conclude that $\Gamma^1_1(S)$ is connected with finite diameter. This will be useful in the proof of connectedness of $\Gamma_k(S)$.

\section{Connectedness and Simple Connectedness of $\Gamma_k(S)$}\label{con}
In this section we prove Theorem \ref{thm:simply}.
\begin{lemma}\label{lem:common}
  Let $v,w$ be cut systems in $\Gamma_k(S)$ so that $v$ and $w$ have all but one curve in common. Then there is a path of length at most $4$ from $v$ to $w$ in $\Gamma^1_1(S)$.
\end{lemma}
\begin{proof}
  Let $v-w = \{a\}$ and $w-v=\{b\}$ and let $R$ be the surface obtained by cutting $S$ along the representatives of all the common curves, $c_1,\cdots, c_{k-1}$, of $v$ and $w$. There is a path $a\lr d_1\lr\cdots\lr d_{n}\lr b$ in $\Gamma^1_1(R) = \mathcal{G}(R)$ with $n\leq 3$. Since $d_i$ are non-separating curves in $R$ it follows that $v_i = (d_i, c_1,\cdots, c_{k-1})$ is a cut system. Thus, $v\lr v_1\lr\cdots \lr v_n\lr w$ is the desired path.
\end{proof}
\begin{proof}[Proof of first half of Theorem \ref{thm:simply}]
  Let $v,w\in \Gamma^0_k(S)$. We use induction on the number of curves $\ell = |v-v\cap w|$. Lemma \ref{lem:common} shows that for the base case $\ell=1$ there is a path from $v$ to $w$. Suppose now that $v-w = \{a_0,\cdots, a_{\ell-1}\}$, $w-v=\{b_0,\cdots, b_{\ell-1}\}$, and $v\cap w = \{c_0,\cdots, c_{k-\ell-1}\}$. Let $K\subset S$ be a compact sub-surface containing all the curves in $v$ and $w$ and let $d$ be a non-separating curve in some component of $S-K$. Then, using induction, there is a path from $v' = (d,a_1,\cdots,a_{\ell-1},c_0,\cdots,c_{k-\ell-1})$ to $w'=(d,b_1,\cdots,b_{\ell-1},c_0,\cdots,c_{k-\ell-1})$. Using Lemma \ref{lem:common}, there is a path from $v$ to $v'$ and $w$ to $w'$.
\begin{figure}[h]
\[\begin{tikzcd}
	v && w \\
	\\
	{v'} && {w'} & {}
	\arrow[squiggly, tail reversed, from=1-1, to=3-1]
	\arrow[squiggly, tail reversed, from=1-3, to=3-3]
	\arrow[squiggly, tail reversed, from=3-1, to=3-3]
\end{tikzcd}\]
\caption{Constructing the path from $v$ to $w$.}
\end{figure}
Using the construction above and the fact that the length of the path in Lemma \ref{lem:common} is at most $4$, it follows that the diameter of $\Gamma^1_k(S)$ is at most $8k-4$.
\end{proof}
If $v,w$ are two cut systems in $\Gamma_k(S)$ so that all of the curves in $v\cup w$ are distinct then any path in $\Gamma^1_k(S)$ between $v$ and $w$ has length at least $k$. This means that the diameter of $\Gamma^1_k(S)$ has a lower bound $k$.\\

In order to prove that $\Gamma_k(S)$ is simply-connected we define the following notion of radius of a path, and segments.
\begin{definition}
  The radius of a path $p = v_0 \lr \cdots \lr v_n$ in $\Gamma^1_k(S)$ with respect to a non-separating curve, $a\in v_j$ for some $j$, is defined as $\max_{v_i}\l(\min_{b\in v_i}i(a,b)\r)$. If $a_0,\cdots, a_n$ are non-separating curves in $S$ then $p$ is a $a_0,\cdots,\ a_n-$segment if $a_0,\cdots,a_n\in v_i$ for each vertex $v_i$ in the path $p$.
\end{definition}

\begin{proposition}\label{prop:base-case}
  Every closed path $p=a_0\lr\cdots\lr a_{n-1}\lr a_0$ in $\Gamma_1(S)$ is null-homotopic.
\end{proposition}

\label{pre}
\textbf{Ends of a Surface.} Let $S$ be a surface of infinite-type and let $\{K_i\}_{i\in \N}$ be a strictly increasing chain of compact sets ordered by inclusion which covers $S$. Then an

\begin{ntheorem}\label{thm:geometric}
  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\text{Aut}(\Gamma_k(S))$ is isomorphic to $\map^*(S)$ for each $k\in \N$.
\end{ntheorem}

\begin{ntheorem}\label{thm:simply}
  Let $S$ be a surface of infinite-type with non-planar ends. Then $\Gamma_k(S)$ is connected and simply connected for each $k\in \N$.
\end{ntheorem}

\begin{ncorollary}\label{cor:iso}
  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\vp:\Gamma_k(S)\to \Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.
\end{ncorollary}

\label{intro}
\addcontentsline{toc}{section}{Introduction}
All surfaces in this paper will be connected, orientable, without any punctures or marked points, and possibly with non-empty compact boundary. The mapping class group, $\text{Map}(S)$, is the group of all orientation preserving homeomorphisms of the surface $S$ up to isotopy. The extended mapping class group, $\text{Map}^*(S)$ is the group of all homeomorphisms of $S$ up to isotopy. When the fundamental group $\pi_1(S)$ of the surface $S$ is finitely generated, we say that $S$ is of finite-type; otherwise $S$ is infinite-type.\\

Recently, there has been an effort in the literature to generalize results about curve graphs and complexes from the finite-type case to the infinite-type case. In \cite{bavard}, the authors show that the automorphism group of the curve complex, $ \mathcal{C}(S)$, of an infinite-type surface with empty boundary is geometric, that is, every automorphism of $ \mathcal{C}(S)$ is induced by the action of $\text{Map}^*(S)$, generalizing the result of \cites{ivanov, luo1999, korkmaz} for finite-type surfaces. Independently, \cite{hernandez} also proved that the automorphism groups of the curve complex, non-separating curve complex, and the Schmutz graph are all isomorphic to each other and to $\text{Map}^*(S)$ for surfaces of infinite-type with no planar ends and finitely many boundary components. Moreover, \cite{hernandez, bavard} show that these complexes can differentiate infinite-type surfaces. In \cite{branman}, the author generalizes the result that automorphism group of the pants complex is isomorphic to the extended mapping class group, a result originally proved in \cite{margalit04} for finite-type surfaces. \\

In this paper, we generalize the Hatcher--Thurston complex to the setting of infinite-type surfaces with possible compact boundary and non-planar ends. The Hatcher--Thurston complex or Cut-system complex, $ \mathcal{HT}(S)$, is a 2-dimensional CW-complex associated to surfaces of finite-type in which the vertices are maximal cut-systems of the surface, edges correspond to an elementary move between cut-systems, and 2-cells are attached to triangles, rectangles, and pentagons as described in Section \ref{pre}. This cut-system complex was first introduced in \cite{hatcher} to obtain a presentation for the mapping class group of finite-type surfaces. In \cite{hatcher}, the authors use a Morse-theoretic proof to show that the complex is connected and simply connected. Later, Wajnryb gave a combinatorial proof of this in \cite{wajnryb}. In \cite{irmak2004}, the automorphisms of the Hatcher--Thurston complex were shown to be geometric. In this paper, the complex $ \mathcal{HT}(S)$ is generalized to a countable collection of complexes $\left(\Gamma_k(S)\right)_{k\in \mathbb{N}}$ whose vertices are given by cut-systems with exactly $k$ non-separating curves, as opposed to maximal cut-systems, with $1-$cells and $2-$cells defined in a similar fashion to $ \mathcal{HT}(S)$. In case that $S$ is a finite-type surface of genus $g$, the collection $\{\Gamma_k(S)\}$ is finite with $1\leq k \leq g$ and $\Gamma_g(S) = \mathcal{HT}(S)$. Precise definitions of these complexes are given in Section \ref{pre}. Since the construction of $1$-cells and $2-$cells of the pants complex in \cite{branman} is similar to that of the Hatcher-Thurston complex, generalization of the pants complex to infinite-type surfaces leads to similar issues with connectedness as the generalization of the Hatcher-Thurston complex. In \cite{branman} these issues are resolved by introducing a family of new topologies on the vertex space and extending it to the enitre pants space. As will be discussed in Section \ref{pre}, we resolve this issue, among others, by instead defining the sequence $(\Gamma_k(S))_{k\in \mathbb{N}}$ of complexes discussed above. In Section \ref{con}, we show the following result.

\textbf{Remark.} At multiple points throughout the proof of Theorem \ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\Gamma_k(S)$ simply connected for all $1\leq k \leq g$ when $S$ is a genus $g$ surface? Since $\Gamma_g(S) = \mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\Gamma_g(S)$ is simply-connected. But the proof of this, in \cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\

Now we describe the construction of the curves $\b_i$. Let $R$ be a subsurface of $S$ with compact boundary and at least one non-planar end so that all the curves $\a_0,\cdots, \a_{n-1}$ are contained in $R$. Let $\delta$ be a separating curve in $R$ disjoint from all $\a_i$ so that the two surfaces, $R_1$ and $R_2$, obtained by cutting $R$ along $\delta$ satisfy the following: $R_1$ is compact and contains all $\a_i$ and $R_2$ is of infinite-type with a single compact boundary component corresponding to the curve $\delta$ in $R$ and has non-planar ends. Let $\tau_0$ and $\tau_1$ be homotopically distinct arcs in $R_2$ with their end points lying on the boundary of $R_2$ so that $\tau_0\cap \tau_1 = \emptyset$. Since $\delta$ and $\a_0$ are disjoint and $i(a_0,a_1)=1$, it can be shown that there exist disjoint essential arcs, $\sigma_0,\ \sigma_1$, with end points on the boundary of $R_1$ corresponding to $\delta$ and $|\sigma_0\cap \a_0| = 1$ and $|\sigma_1\cap \a_1|=1$. Gluing $R_1$ and $R_2$ together along the boundary corresponding to $\delta$ in such a way that $\beta_0 = \sigma_0\cup \tau_0$ and $\beta_1 = \sigma_1\cup \tau_1$ are closed curves. Then by construction we have $i(b_0,a_0) = i(b_1,a_1) = 1$ and that $b_i$ are non-separating. Let $\b_2$ be a non-separating curve in $R_2$ which intersects $\tau_0$ and $\tau_1$ exactly once. Thus we have our desired curves $b_0,\ b_1,$ and $b_2$.
\end{proof}
The above proposition implies that $\Gamma_1(S)$ is simply-connected when $S$ has non-planar ends. We use this as our base case for induction on $k$, to show that $\Gamma_k(S)$ is also simply-connected. From here on we assume the induction hypothesis: for every surface $S$ the complex $\Gamma_j(S)$ is simply connected for all $j<k$. We prove the simple connectedness now by also inducting on radius of the path. We start by a special case of radius 0 curves.
\begin{proposition}\label{prop:sp-radius-0}
 Suppose that $p = v_0 \lr \cdots \lr v_{n-1}$ is a closed path in $\Gamma_k(S)$ so that all the vertices $v_i$ contain a common curve $c$. Then $p$ is null-homotopic.
\end{proposition}
\begin{proof}
 Let $p'$ be the path in $\Gamma_{k-1}(S)$ where the vertices are $v_i- \{c\}$. By the induction hypothesis, it follows that $p'$ is null-homotopic, meaning that it can be split into paths of type 1, 2, and 3. Since adding $c$ back to each vertex in $p'$ does not remove any edge connections therefore $p$ is null-homotopic.
\end{proof}
\begin{proposition}[Lemma 13, \cite{wajnryb}]\label{prop:hex-path}
 Let $a_0, a_1, a_2$ be disjoint non-separating curves so that:
 \begin{enumerate}
 \item all unions $\a_i\cup \a_j$ are non-separating in $S$.
 \item the union $\a_0\cup \a_1\cup \a_2$ separates $S$.
 \end{enumerate}
 Then there exists disjoint non-separating curves $b_0, b_1, b_2$ so that $i(a_i,b_i) = 1$, $i(a_{i+1}, b_{i}) = 1$ (indices modulo $3$), and $i(a_i, b_{i+1}) = 0$ for all other $i,j$. Moreover, if $(a_0,a_1,c_2,\cdots, c_{k-1})$ is a vertex in $\Gamma_k(S)$ then the closed path $p = v_0 \lr \cdots \lr v_5 \lr v_0$ defined as
 \begin{align*}
 (a_0, a_1, c_2,\cdots, c_{k-1})\lr &(a_0, b_1,c_2,\cdots, c_{k-1}) \lr (a_0, a_2,c_2,\cdots, c_{k-1}) \lr (b_0, a_2,c_2,\cdots, c_{k-1})\\
 &\ \lr (a_1,a_2,c_2,\cdots, c_{k-1})\lr (a_1, b_2,c_2,\cdots, c_{k-1})\lr (a_0, a_1,c_2,\cdots, c_{k-1})
 \end{align*}
 is null-homotopic in $\Gamma_k(S)$.
\end{proposition}
\begin{proof}
 Due to conditions on $a_0,\ a_1,\ a_2$ it follows that $S - \bigcup_{i}\a_i$ has exactly two connected components, $R_1$ and $R_2$, both with at least three boundary components corresponding to the curves $\a_0,\ \a_1,$ and $\a_2$. Let $\delta_i$ be disjoint arcs in $R_1$ with endpoints on the boundary components corresponding to $\a_i$ and $\a_{i+1}$ (indices taken modulo 3). Similarly define disjoint arcs $\delta'_i$ on $R_2$. Defining $\b_i = \delta_i\cup \delta'_i$ we get disjoint non-separating curves $b_0, b_1, b_2$ with the desired intersections with $a_i$. In order to prove the second part of the proposition, define $c = T_{a_1}(b_0)$. Then it follows from Proposition \ref{prop:dehn-int} that $i(c, b_0) = i(c, a_1) = i(c,b_1) = i(c,a_0) = 1$ and $i(c,b_2) = i(c, a_2) = 0$. If $(a_0, a_1, c_2,\cdots, c_{k-1})$ is a vertex in $\Gamma_k(S)$ then define the vertices $w_0 = (b_1,b_2, c_2,\cdots, c_{k-1}), w_1 = (c,b_2,c_2,\cdots, c_{k-1}), w_2 = (c,a_2,c_2,\cdots, c_{k-1})$. Then the path $p = v_0 \lr \cdots \lr v_5 \lr v_0$ as defined above can be split into paths of type 1, 2, and 3 as in Figure \ref{fig:hex-path}.
 \begin{figure}
 \[\begin{tikzcd}
 {v_0} && {v_1} && {v_2} && {v_3} \\
 \\
 && {w_0} && {w_2} \\
 &&& {w_1} \\
 {v_5} &&&&&& {v_4}
 \arrow[tail reversed, from=1-1, to=1-3]
 \arrow[tail reversed, from=1-3, to=1-5]
 \arrow[tail reversed, from=1-5, to=1-7]
 \arrow[tail reversed, from=1-7, to=5-7]
 \arrow[tail reversed, from=3-3, to=1-3]
 \arrow[tail reversed, from=3-3, to=4-4]
 \arrow[tail reversed, from=3-3, to=5-1]
 \arrow[tail reversed, from=3-5, to=1-5]
 \arrow[tail reversed, from=3-5, to=1-7]
 \arrow[tail reversed, from=3-5, to=5-7]
 \arrow[tail reversed, from=4-4, to=3-5]
 \arrow[tail reversed, from=4-4, to=5-1]
 \arrow[tail reversed, from=5-1, to=1-1]
 \arrow[tail reversed, from=5-7, to=5-1]
\end{tikzcd}\]
 \caption{Splitting the path $p$ into paths of type $1$, $2$, and $3$.}
 \label{fig:hex-path}
 \end{figure}
\end{proof}
Before beginning the proof of the next lemma, note that if $v$ and $w$ are vertices in $\Gamma_k(S)$ with common curves $(a_0, \cdots, a_m)$ then there is an $(a_0,\cdots, a_m)-$segment connecting $v$ and $w$. This follows directly from the proof of connectedness. This fact will be repeatedly used in the proof of the next lemma.
\begin{lemma}\label{lem:radius-zero}
 Let $p$ be a path in $\Gamma_k(S)$ of radius $0$ with respect to some curve $a_0$. Then $p$ is null-homotopic.
\end{lemma}
\begin{proof}
 We prove this by induction on the number of segments in the path $p$. Suppose that $a_0\in v_0$, where $v_0$ is a vertex in $p$. We call the maximal $a_0-$segment containing $v_0$ the first segment. From Proposition \ref{prop:sp-radius-0} it follows that if $p$ is a $a_0-$segment then $p$ is null homotopic. Suppose this is not the case and $v_1$ is the terminal vertex of the first segment. Let $a_1\in v_1$ be a curve distinct from $a_0$ which will serve as the common curve of the next segment, which we call the second segment. Let $v_2$ be the terminal element of second segment. If $p$ is just made up of these two segments then both $v_1$ and $v_2$ contain $a_0$ and $a_1$, thus there is a $(a_0, a_1)-$segment from $v_1$ to $v_2$, therefore implying that $p$ can be split into a $a_0-$segment and a $a_1-$segment, both of which are null-homotopic due to Proposition \ref{prop:sp-radius-0}. Suppose now that $p$ has more than $2$ segments. Then let $a_2$ be a curve in the vertex $v$ which follows $v_2$, so that $i(a_0, a_2)=0$. Let $v_3$ be the terminal vertex in the $a_2-$segment. Now we try to reduce the number of segments.\\

The proof of this result closely follows the combinatorial proof of \cite{wajnryb} with alterations made when required for the infinite-type case. The proof proceeds by induction on $k$. We first show that the $1-$skeleton, $\Gamma^1_k(S)$, is connected and that its diameter is bounded between $k$ and $8k-4$ using induction on $k$. To further prove that $\Gamma_k(S)$ is simply connected, we introduce the notions of radius of a path in $\Gamma_k(S)$ and segments, following the proof by Wajnryb. The remaining proof can then be divided into three steps. It is first shown in Proposition \ref{prop:base-case} that $\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible. Note that this argument uses the assumption that $S$ has at least one non-planar end. Continuing the induction on $k$, the second step shows that any closed 0-radius path is contractible in $\Gamma_k(S)$, see Proposition \ref{lem:radius-zero}, by induction on the number of segments in the path. Finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.\\

\textbf{Remark.} At multiple points throughout the proof of Theorem \ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\Gamma_k(S)$ simply connected for all $1\leq k \leq g$ when $S$ is a genus $g$ surface? Since $\Gamma_g(S) = \mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\Gamma_g(S)$ is simply-connected. But the proof of this, in \cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\

In Section \ref{aut}, we show that automorphisms of $\Gamma_k(S)$ are geometric:
\begin{ntheorem}\label{thm:geometric}
  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\text{Aut}(\Gamma_k(S))$ is isomorphic to $\text{Map}^*(S)$ for each $k\in \mathbb{N}$.
\end{ntheorem}
We prove this by constructing an action of $\text{Aut}(\Gamma_k(S))$ on the Schmutz graph, $ \mathcal{G}(S)$, as done by \cite{irmak2004}. The proof in \cite{irmak2004} is almost entirely transferable to the infinite-type case, but parts of it are reproduced here for the sake of completeness. As a direct corollary to Theorem \ref{thm:geometric} we have the following:
\begin{ncorollary}\label{cor:iso}
  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\varphi:\Gamma_k(S)\to \Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.
\end{ncorollary}
The Hatcher--Thurston complex is an important tool in the study of mapping class group of finite-type surfaces. By studying the stabilizers of vertices, edges, and cells in $ \mathcal{HT}(S)$ of the action of $\text{Map}(S)$ Wajnryb gave a simple presentation of $\text{Map}(S)$ in \cite{wajnryb2}, which was inspired by a similar approach to the problem by Harer in \cite{harer}. Harer in the same paper uses a simpler version of the Hatcher--Thurston complex to compute the second cohomology of the mapping class group. These methods are not directly transferable to big mapping class group as they are not finitely generated, in fact they are not even countably generated. In \cite{patel}, the authors give topological generators for the pure mapping class group $\text{PMap}(S)$, the subgroup of $\text{Map}(S)$ which are end-preserving, in terms of Dehn Twists and handle shift maps. For infinite-type surfaces with countably many non-planar ends it may be possible to use the results presented in this paper to determine relations in $\text{PMap}(S)$ between Dehn Twists and handle shift maps by applying the techniques of Wajnryb and Harer.
\vspace{1em}
\newline
\textbf{Outline.} In Section \ref{pre} the main objects of this paper are defined and conventions and notations which are used throughout the paper are discussed. Section \ref{con} is dedicated to the proof of Theorem \ref{thm:simply} and Section \ref{aut} to the proof of Theorem \ref{thm:geometric} and Corollary \ref{cor:iso}.
\vspace{1em}
\newline
\textbf{Acknowledgements.} The author would like to thank Sumanta Das for posing the question which led to this paper and his comments for improving the quality of the paper. The author also extends gratitude to Dan Margalit and Tyrone Ghaswala, whose comments helped improve the paper.

The proof of Theorem~\ref{thm:simply} “closely follows the combinatorial proof of \cite{wajnryb} with alterations made when required for the infinite-type case” and “proceeds by induction on $k$.” First, one shows “that the $1$-skeleton, $\Gamma^1_k(S)$, is connected” and that “its diameter is bounded between $k$ and $8k-4$ using induction on $k$.” For simple connectedness, the proof “introduce[s] the notions of radius of a path in $\Gamma_k(S)$ and segments, following the proof by Wajnryb,” and then splits into three steps: (1) in Proposition~\ref{prop:base-case}, prove “$\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible,” using the assumption that “$S$ has at least one non-planar end”; (2) for general $k$, show “that any closed $0$-radius path is contractible in $\Gamma_k(S)$, see Proposition~\ref{lem:radius-zero}, by induction on the number of segments in the path”; (3) “finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.”

Question: Let $S$ be a connected, orientable surface with no punctures or marked points and possibly nonempty compact boundary. Assume that $S$ is of infinite type, meaning that $\pi_1(S)$ is not finitely generated, and that $S$ has non-planar ends. For each $k\in\mathbb{N}$, let $\Gamma_k(S)$ denote the generalized Hatcher--Thurston $2$-dimensional CW-complex whose vertices are cut-systems consisting of exactly $k$ non-separating curves on $S$, with $1$-cells and $2$-cells defined analogously to the Hatcher--Thurston complex. Which statement holds for every such surface $S$ and every $k\in\mathbb{N}$?

For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected and simply connected.

Choices:
B. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected, and it is simply connected whenever $k$ is sufficiently large.
C. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected.
D. There exists $N\in\mathbb{N}$ such that for every $k\ge N$, the complex $\Gamma_k(S)$ is connected and simply connected.
E. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is path connected and contractible.

simple-connectedness strengthened to contractibility