The relationship between the geometry of manifolds and the analytic properties of functions on manifolds is a defining theme of geometric analysis. Our direction starts with the Liouville theorems for harmonic functions on \(\mathbb{R}^n\) and Yau's generalization.

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

A natural generalization is to try to show this result for solutions of the heat equation. However, the heat equation is very flexible compared to the Laplace equation, and since there are bounded solutions to the heat equation, no Liouville theorem is possible in general.

Despite this, if we restrict attention to specifically \textit{ancient} solutions of the heat equation, that is, solutions which are defined for all time going back to \(-\infty\), then Liouville theorems actually do become possible. Indeed, in \cite{colding2021optimal}, Colding and Minicozzi generalize \cite{colding1997harmonic} to show that the space \(\mathcal{P}_d(M)\) of ancient solutions of the heat equation with polynomially bounded growth also has finite dimension. In (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}), Colding and Minicozzi show how the spaces \(\mathcal{P}_d(M)\) are relevant to geometric flows.

Continuing to more types of equations, Wang and Zhu recently generalized the result of \cite{colding1997harmonic} to biharmonic functions \cite{wang2025qualitativebehaviorbiharmonicfunctions}, i.e., functions \(u: M \rightarrow \mathbb{R}\) solving
\begin{equation*}
    \Delta\Delta u = 0.
\end{equation*}

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
    \begin{equation*}
        \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
    \end{equation*}
    such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
    \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
    \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

Yau proved in \cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \(M\) with nonnegative Ricci curvature, the space \(\mathcal{H}_d(M)\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \cite{colding1997harmonic}.

This equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.

Our goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \cite{colding2021optimal}. Our main result is
\begin{theorem*}
 Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(u: M \times (-\infty, 0] \rightarrow \mathbb{R}\) be an ancient solution of
 \begin{equation*}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0
 \end{equation*}
 such that \(|u(x, t)|\) and \(|\nabla u(x, t)|\) have polynomially bounded growth in the heat balls \(B_R(x) \times [-R^4, 0]\). The space of all such solutions \(u(x, t)\) is finite dimensional.
\end{theorem*}

\subsection{Definitions and Notation}
We now give more precise definitions and statements. Given a manifold \(M\) and an interval \(I \subset \mathbb{R}\), a function \(u: M \times I \rightarrow \mathbb{R}\) satisfies the biharmonic heat equation if
\begin{equation}\label{first instance of biharmonic heat equation}
 \partial_t u(x, t) + \Delta\Delta u(x, t) = 0.
\end{equation}
We will call such a function ``bicaloric'' for brevity. A bicaloric function \(u\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \(t \in (-\infty, 0]\). We say that \(u \in \mathcal{P}_{d, d'}(M)\) for \(d, d' > 0\) if \(\partial_t u + \Delta\Delta u = 0\), \(u\) is ancient, and for some constants \(C, C' > 0\),
\begin{equation}\label{definition of polynomially bounded growth in introduction}
 \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}
for any \(p \in M\) and \(R > 0\). We similarly say that \(u \in \mathcal{H}_{d, d'}(M)\) if \(\Delta\Delta u = 0\) and the same bounds in (\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \(B_R(p)\).

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

With these definitions, our main results are more precisely stated as

\begin{equation}\label{definition of polynomially bounded growth in introduction}
    \sup_{B_R(p) \times [-R^4, 0]}|u(x, t)| \le C(1 + R)^d, \quad \sup_{B_R(p) \times [-R^4, 0]}|\nabla u(x, t)| \le C'(1 + R)^{d'}
\end{equation}

A manifold \(M\) is said to have polynomial volume growth if there are constants \(C, d_V > 0\) and some \(p \in M\) such that \(\operatorname{Vol}(B_R(p)) \le C(1 + R)^{d_V}\) for all \(R> 0\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \(K\) if for some \(p \in M\) and all \(R > 0\),
\begin{equation}
 \sup_{v \in TB_{R}(p)}\frac{\Ric(v, v)}{|v|^2} \ge -\frac{K}{R^2}.
\end{equation}

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
 Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

Since there are \(d + 1\) vectors \((1, t_i, \ldots, t_i^{d + 1})\), they span \(\mathbb{R}^{d + 1}\), and so there are constants \(b_i^j\) such that
 \begin{equation}\label{bdefinitions}
 e_j = b_i^j(1, t_i, \ldots, t_i^{d}).
 \end{equation}
 It now follows that
 \begin{equation}\label{expression of p in b}
 p_j(x) = b_i^ju(x, t_i), \quad \nabla p_j(x) = b_i^j\nabla u(x, t_i),
 \end{equation}
 and we conclude that \(p_j\) can grow at most polynomially of degree \(4k\) and \(\nabla p_j\) can grow at most polynomially of degree \(4\ell\). Because \(p_j\) vanishes when \(j > k\) and \(\nabla p_j\) vanishes when \(j > \ell + 1\), we have
 \begin{equation}
 u = p_0 + t p_1 + \ldots + t^{k}p_{k} \quad \text{and} \quad \nabla u = \nabla p_0 + t\nabla p_1 + \ldots + t^{\ell}\nabla p_{\ell},
 \end{equation}
 it follows that
 \begin{equation}
 \begin{split}
 |u(x, t)| \le C(1 + |t|^k + |x|^{4k}) \quad \text{and} \quad |\nabla u(x, t)| \le C(1 + |t|^{\ell} + |x|^{4\ell})
 \end{split}
 \end{equation}
 From equation (\ref{bdefinitions}) we have
 \begin{equation}
 \begin{split}
 \sum_i b_i^ju(x, R^4t_i) & = \sum_i\sum_{m}b_i^jp_{m}(x)R^{4j}t_i^{m} = R^{4j}\sum_mp_m(x)\left(\sum_ib_{i}^jt_i^m\right) \\ & = \sum R^{4j}\sum_m p_m(x)\delta_{mj} \\ & = R^{4j}p_j(x).
 \end{split}
 \end{equation}
 Similarly,
 \begin{equation}
 \sum_i b_i^j\nabla u(x, R^4t_i) = R^{4j}\nabla p_j(x)
 \end{equation}
 Thus
 \begin{equation}
 \begin{split}
 |R^{4j}p_j(x)| & = \left|\sum_i b_i^ju(x, R^4t_i)\right| \le A\sum_i\left|u(x, R^4t_i)\right| \\ & \le A(1 + |x|^{4k} + \sum_i|Rt_i|^{4d}) \le AR^{4k}
 \end{split}
 \end{equation}
 and similarly
 \begin{equation}
 \begin{split}
 |R^{4j}\nabla p_j(x)| & \le A'(1 + |x|^{4\ell} + \sum_i|Rt_i|^{4\ell}) \le A'R^{4\ell},
 \end{split}
 \end{equation}
 so that \(|p_j(x)| \le A_jR^{4(k - j)}\) and \(|p_j(x)| \le A_j'R^{4(\ell - j)}\).
\end{proof}
\begin{proof}[Proof of Theorem \ref{dimension bound theorem}]
 Choose some \(u \in {P}_{4k, 4\ell}(M)\) and suppose \(u = p_0(x) + tp_1(x) + \cdots + t^dp_d(x)\), as in lemma \ref{coefficientbound}. Then \(\Delta\Delta p_{d} = 0\) and for \(j < d\), \(\Delta\Delta p_j = -(j+1)p_{j + 1}\). Thus, there a linear map \(\Psi_0: \mathcal{P}_{4k, 4\ell} \rightarrow \mathcal{H}_{4k, 4\ell}\) defined by \(\Psi_0u = p_{d}\) (here we use the coefficient estimate in (\ref{coefficients in polynomial expression bounds})). If we let \(\mathcal{K}_0 = \ker \Psi_0\) we find
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} \le \dim \mathcal{K}_0 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 If \(u \in \mathcal{K}_0\), then \(p_{d} = 0\) and \(\Delta\Delta p_{d - 1} = -dp_{d} = 0\), and so we have a map \(\Psi_1: \mathcal{K}_0 \rightarrow \mathcal{H}_{4(k - 1), 4(\ell - 1)}\) defined by setting \(\Psi_1u = p_{d - 1}\). Letting \(\ker \Psi_1 = \mathcal{K}_1\) then
 \begin{equation}
 \dim \mathcal{K}_0 \le \dim \mathcal{K}_1 + \dim \mathcal{H}_{4k, 4\ell}
 \end{equation}
 When \(k \le \ell + 1\), we can repeat this \(k\) times to get
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell}(M) \le \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M)
 \end{equation}
 When \(k > \ell + 1\), we have from lemma \ref{coefficientbound} that \(\nabla p_{\ell + 1} = 0\), and so \(p_{\ell + 1} = p_d\) is a constant. Thus in this case \(p_d\) lies in a one dimensional subspace of \(H_{4k, 4\ell}(M)\), so that
 \begin{equation}
 \dim \mathcal{P}_{4k, 4\ell} = \dim \mathcal{K}_0 + 1.
 \end{equation}
 We then iterate the same argument as before, \(\ell\) times, to get the second inequality in (\ref{equation in the first statement of the main theorem}).

\end{proof}
\subsection{Bicaloric Polynomials in \(\mathbb{R}^n\)}
Now we consider polynomially bounded solutions to the biharmonic heat equation in \(\mathbb{R}^n\).
\begin{proposition}\label{bicaloric functions in Rn are polynomials}
 Let \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\). Then \(u\) is a polynomial in \(x_i\) and \(t\).
\end{proposition}
\begin{proof}
 As before, this follows from the reverse Poincaré estimate and the fact that the operators \(\partial_{x_i}\), \(\partial_t\), and \(\Delta\) commute in \(\mathbb{R}^n\).
\end{proof}
\begin{corollary}
 If \(u \in \mathcal{P}_{k, \ell}(\mathbb{R}^n)\), then there is some \(d\) such that \(u \in \mathcal{P}_{d, d-1}(\mathbb{R}^n)\).
\end{corollary}\label{fewer caloric spaces in R^n}
\begin{proof}
 This follows the same way as before.
\end{proof}
Given a monomial in \(x_i\) and \(t\), we define its biparabolic degree as follows: for \(t^{n_0}\prod x_i^{n_i}\), the biparabolic degree is \(4n_0 + \sum n_i\). The degree of a polynomial is then the maximal degree of the monomials summing to it. Let \(\mathcal{A}_j^n\) be the set of homogeneous polynomials in \(\mathbb{R}^n\) of biparabolic degree \(j\). We have
\begin{equation}
 \mathcal{A}_d^n = A_d^n \oplus t A_{d - 4}^n \oplus t^2A_{d - 8}^n \oplus \cdots
\end{equation}
\begin{lemma}\label{dim for bicaloric polynomials}
 For \(d > 0\) we have \(\dim(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim A_d^n\) and
 \begin{equation}
 \dim \mathcal{P}_{d, d-1}(\mathbb{R}^n) = \sum_{j = 0}^d\dim A_j^n.
 \end{equation}
\end{lemma}
\begin{proof}
 Both \(\partial_t\) and \(\Delta\Delta\) map \(\mathcal{A}_d^n\) to \(\mathcal{A}_{d-4}^n\). We note that for \(u \in \mathcal{A}_{d-4}^n\) we have
 \begin{equation}
 (\partial_t + \Delta\Delta)\left(tu - \frac{1}{2}t^2(\partial_t + \Delta\Delta)u + \frac{1}{6}t^3(\partial_t + \Delta\Delta)^2 u - \cdots\right) = u,
 \end{equation}
 so that \(\partial_t + \Delta\Delta\) is surjective. Thus,
 \begin{equation}
 \dim (\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n) = \dim \mathcal{A}_d^n - \dim\mathcal{A}_{d-4}^n = \dim A_d^n
 \end{equation}
 since \(\mathcal{P}_{d, d-1}(\mathbb{R}^n) \cap \mathcal{A}_d^n\) is the kernel of \(\partial_t + \Delta\Delta\) restricted to \(\mathcal{A}_d^n\). Summing gives the second claim.

\end{proof}
Now finally we show that the estimate in Theorem \ref{dimension bound theorem} is sharp in \(\mathbb{R}^n\).
\begin{corollary}\label{sharp in Rn}
 For positive integers \(d > 0\)
 \begin{equation}
 \dim \mathcal{P}_{4d, 4d - 1}(\mathbb{R}^n) = \sum_{i = 0}^{d} \dim \mathcal{H}_{4(d - i), 4(d - i) - 1}(\mathbb{R}^n).
 \end{equation}
\end{corollary}
\begin{proof}
 From lemmas \ref{dim for bicaloric polynomials} and \ref{dim for biharmonic polynomials} we have
 \begin{equation}
 \begin{split}
 \dim \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n) & = \sum_{j = 0}^{4d}\dim A_j^n = \sum_{j = 0}^d(\dim A_{4j}^n + \dim A_{4j - 1}^n + \dim A_{4j - 2}^n + \dim A_{4j - 3}^n) \\ & = \sum_{j = 0}^d\dim \mathcal{H}_{4d, 4d - 1}(\mathbb{R}^n).
 \end{split}
 \end{equation}
 Setting \(k = \ell = d\) in Theorem \ref{dimension bound theorem} and noting \(\mathcal{P}_{4d, 4d}(\mathbb{R}^n) = \mathcal{P}_{4d, 4d-1}(\mathbb{R}^n)\) and \(\mathcal{H}_{4d, 4d}(\mathbb{R}^n) = \mathcal{H}_{4d, 4d-1}(\mathbb{R}^n)\) shows that the estimate in Theorem \ref{dimension bound theorem} is sharp.

Combining this with Wang and Zhu's result \cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:

\begin{corollary}
    Let \(M\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \(k, \ell \ge 0\) the spaces \(\mathcal{P}_{4k, 4\ell}(M)\) are finite dimensional.
\end{corollary}

\subsection{Harmonic and biharmonic functions}

Biharmonic functions arise in several variational problems. Just as minimizing \(\int|\nabla u|^2\) leads one to the Laplace and heat equations, minimizing \(\int |\Delta u|^2\) leads to the biharmonic and biharmonic heat equations.

In general, fourth order elliptic operators arise naturally when taking variations involving second order objects, one major example being variations of metrics in conformal geometry (see \cite{chang1995extremal}, \cite{Lin1998classification}). They also arise in the study of the Willmore energy. For an immersed surface \(\phi: M^2 \rightarrow \mathbb{R}^3\), the Willmore energy is defined as
\begin{equation}\label{Willmore functional}
    \mathcal{W}(\phi) = \int_{M}H^2\,dA
\end{equation}
where \(dA\) is the induced volume element and \(H\) is the mean curvature \cite{willmore2000surfaces}. In studying critical points of this functional one arrives at the Euler-Lagrange equation
\begin{equation}\label{euler-lagrange equation}
    \Delta H + 2H(H^2 - K) = 0,
\end{equation}
a fourth order elliptic operator. The biharmonic heat equation similarly arises when studying the gradient flow of the Willmore energy (\cite{kuwert2002gradient}, \cite{lamm2005biharmonic}). Ancient solutions to heat equations often appear when doing blowup analysis of general solutions to a variational problem. See \cite{kuwert2004removable} for blowup analysis of singularities of Willmore flows. We also again reference (\cite{colding2020complexity}, \cite{colding2019search}, \cite{colding2019liouville}) for more on how ancient solutions to heat equations with polynomially bounded growth are relevant to geometric flows.

Although both arise from variational problems, biharmonic functions in general differ significantly from harmonic functions, because no maximum principle holds for biharmonic functions. This limits the kinds of estimates we can find for biharmonic functions. In particular, the usual pointwise derivative estimates one can find for harmonic functions on a ball cannot be found for a biharmonic function.

On the bright side, energy methods for harmonic and caloric functions seem to have analogs for biharmonic and bicaloric functions, which we will see as we prove Theorem \ref{dimension bound theorem}. We are still limited to some extent, however, because when performing integrations by parts we are forced to use the Bochner formula
\begin{equation*}
    \frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\end{equation*}
to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\). It is the appearance of the Ricci term here that makes the decay on Ricci curvature crucial for our result.

Our methodology is inspired by Colding and Minicozzi's in \cite{colding2021optimal}. We will show a reverse Poincaré inequality for bicaloric functions on ``heat balls'' \(B_R(p) \times [-R^4, 0]\). Because we are considering ancient bicaloric functions, we will be able to apply the inequality as \(R \rightarrow \infty\) to get strong, global control of their behavior. In particular we will see that high order time derivatives \(\partial_t^ku\) must vanish identically, allowing us to write for some finite \(d\):
\[u(x, t) = p_d(x)t^d + \cdots + p_1(x)t + p_0(x)\]
with \(\Delta \Delta p_d = 0\) and \(\Delta \Delta p_j = -(j + 1)p_{j + 1}\). This will allow us to directly compare the spaces \(\mathcal{H}_{4k, 4\ell}(M)\) with \(\mathcal{P}_{4k, 4\ell}(M)\).

To show the dimension estimates are sharp in \(\mathbb{R}^n\), we will consider biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions of the spaces \(\mathcal{H}_{4k, 4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k, 4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for Theorem~\ref{dimension bound theorem} as follows.

1. **Use energy methods and Bochner to handle fourth order terms.** It says that while working "as we prove Theorem \ref{dimension bound theorem}" one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is "forced to use the Bochner formula"
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
"to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\)," and "the appearance of the Ricci term here" is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will "show a reverse Poincar\'e inequality for bicaloric functions on `heat balls' \(B_R(p)\times[-R^4,0]\)."

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, "we will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior."

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, "high order time derivatives \(\partial_t^k u\) must vanish identically," so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations "\(\Delta\Delta p_d=0\)" and "\(\Delta\Delta p_j=-(j+1)p_{j+1}\)."

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions "will allow us to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\)," yielding the dimension bound.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, it says they will consider "biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions" of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

The introduction indicates the proof strategy for the following theorem.

\begin{theorem}\label{dimension bound theorem}
    Let \(M\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \(k\), \(\ell\) be nonnegative integers. Then
    \begin{equation}\label{equation in the first statement of the main theorem}
        \dim \mathcal{P}_{4k, 4\ell}(M) \le \begin{cases}
             \displaystyle \sum_{i = 0}^{k} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k \le \ell + 1, \\ \displaystyle
            1 + \sum_{i = 0}^{\ell} \dim \mathcal{H}_{4(k - i), 4(\ell - i)}(M) & \quad k > \ell + 1
        \end{cases}
    \end{equation}
    Moreover, these inequalities are sharp in \(\mathbb{R}^n\).
\end{theorem}

1. **Use energy methods and Bochner to handle fourth order terms.** In establishing the main theorem, one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is forced to use the Bochner formula
\[
\frac{1}{2}\Delta|\nabla u|^2 = |\nabla^2u| + \langle \nabla \Delta u, \nabla u\rangle + \Ric(\nabla u, \nabla u)
\]
so as to control the factor \(\langle \nabla \Delta u, \nabla u\rangle\); the appearance of the Ricci term here is why the quadratic lower bound/decay assumption on Ricci is crucial.

2. **Establish a reverse Poincar\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will show a reverse Poincar\'e inequality for bicaloric functions on “heat balls” \(B_R(p)\times[-R^4,0]\).

3. **Send \(R\to\infty\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, one will be able to apply the inequality as \(R\to\infty\) to get strong, global control of their behavior.

4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, high order time derivatives \(\partial_t^k u\) must vanish identically, so for some finite \(d\)
\[
u(x,t)=p_d(x)t^d+\cdots+p_1(x)t+p_0(x),
\]
with relations “\(\Delta\Delta p_d=0\)” and “\(\Delta\Delta p_j=-(j+1)p_{j+1}\).”

5. **Compare \(\mathcal{H}_{4k,4\ell}(M)\) and \(\mathcal{P}_{4k,4\ell}(M)\) via the coefficient relations.** The stated expansion and recursions allow one to directly compare the spaces \(\mathcal{H}_{4k,4\ell}(M)\) with \(\mathcal{P}_{4k,4\ell}(M)\), yielding the dimension bound \eqref{equation in the first statement of the main theorem}.

6. **Sharpness in \(\mathbb{R}^n\) by explicit polynomial models.** To show sharpness, one considers biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling one to explicitly compute the dimensions of \(\mathcal{H}_{4k,4\ell}(\mathbb{R}^n)\) and \(\mathcal{P}_{4k,4\ell}(\mathbb{R}^n)\).

Question: Let \(M\) be a complete Riemannian manifold. Say that \(M\) has polynomial volume growth if there exist \(p\in M\) and constants \(C,d_V>0\) such that \(\operatorname{Vol}(B_R(p))\le C(1+R)^{d_V}\) for all \(R>0\). Say that the Ricci curvature of \(M\) is bounded below quadratically if there exist \(p\in M\) and \(K>0\) such that \(\sup_{v\in TB_R(p)}\frac{\operatorname{Ric}(v,v)}{|v|^2}\ge -\frac{K}{R^2}\) for all \(R>0\). For \(d,d'\), let \(\mathcal{P}_{d,d'}(M)\) denote the space of ancient functions \(u:M\times(-\infty,0]\to\mathbb{R}\) satisfying the biharmonic heat equation \(\partial_t u+\Delta\Delta u=0\) and such that for some constants \(C,C'>0\), \(\sup_{B_R(q)\times[-R^4,0]}|u(x,t)|\le C(1+R)^d\) and \(\sup_{B_R(q)\times[-R^4,0]}|\nabla u(x,t)|\le C'(1+R)^{d'}\) for every \(q\in M\) and \(R>0\). Let \(\mathcal{H}_{d,d'}(M)\) denote the space of functions \(u:M\to\mathbb{R}\) satisfying \(\Delta\Delta u=0\) and the analogous bounds over \(B_R(q)\). For nonnegative integers \(k\) and \(\ell\), which statement holds for every such manifold \(M\)?

\[\dim \mathcal{P}_{4k,4\ell}(M)\le \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell+1,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell+1,\end{cases}\] and moreover these inequalities are sharp in \(\mathbb{R}^n\).

Choices:
B. \[\dim \mathcal{P}_{4k,4\ell}(M)\le \sum_{i=0}^{\min\{k,\ell\}}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M),\] and moreover this inequality is sharp in \(\mathbb{R}^n\).
C. \[\dim \mathcal{P}_{4k,4\ell}(M)<\infty\] for all nonnegative integers \(k\) and \(\ell\).
D. \[\dim \mathcal{P}_{4k,4\ell}(M)\le \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell,\end{cases}\] and moreover these inequalities are sharp in \(\mathbb{R}^n\).
E. \[\dim \mathcal{P}_{4k,4\ell}(M)= \begin{cases}\displaystyle \sum_{i=0}^{k}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k\le \ell+1,\\[4pt]\displaystyle 1+\sum_{i=0}^{\ell}\dim \mathcal{H}_{4(k-i),4(\ell-i)}(M), & k>\ell+1,\end{cases}\] and moreover these equalities are sharp in \(\mathbb{R}^n\).

The stem gives definitions and hypotheses but does not reveal the dimension bound itself. There is no explicit or obvious hint that singles out choice A.

The item is essentially asking for the exact theorem statement under the stated hypotheses. The multiple close alternatives prevent it from being a pure verbatim restatement, but it still mainly tests theorem recall rather than an independently derived conclusion.

Some reasoning is needed to distinguish subtle variants: off-by-one thresholding, inequality versus equality, and the extra constant term in one regime. However, the task is mostly recognition of the correct theorem statement rather than substantial mathematical generation.

The distractors are strong: one is a weaker true statement, others are plausible near-miss versions with boundary errors or overstrong equality claims. They are distinct and reflect realistic failure modes in reading or recalling the result.

A solid theorem-recall MCQ with strong distractors and no answer leakage, but it mainly tests precise recall/discrimination among near-identical statements rather than deep generative reasoning.


Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\min}(M,g)$
the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.
In general, the existence of such a minimal object follows from the
Almgren--Pitts min--max theory \cite{pitts2014existence}, and in low
codimension and low dimension one can often upgrade the varifold to a smooth
embedded minimal hypersurface using the regularity theory of
Schoen--Simon--Yau \cite{schoen1975,schoen1981regularity}.

In this article we are interested in bounding $A_{\min}(M,g)$ in terms of
global geometric data in the special case when $(M,g)$ is an Einstein
$4$-manifold subject to natural non-collapsing and diameter bounds.
Throughout we assume that
\begin{equation}\label{eq:Ein-assumptions}
 \Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad
 \diam(M,g)\le D,\quad H_1(M;\mathbb{Z})=0.
\end{equation}
We write $\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds
satisfying \eqref{eq:Ein-assumptions}.

This paper is a sequel to our previous work \cite{LW_HF1}, where we considered
the larger class with $ |{\rm Ric}_g|\le 3,\ \Vol(M,g)\ge v>0,\
 \diam(M,g)\le D,\ H_1(M;\mathbb{Z})=0, $
and proved a linear upper bound for the first homological filling function
$\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\in\Mm_1(4,v,D)$ we
showed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that
\[
 \HF_1(l)\ \le\ f_1(v,D)\,l + f_2(v,D)\qquad\text{for all }l\ge 0.
\]
Combined with an effective version of the Almgren--Pitts min--max theory due to
Nabutovsky--Rotman \cite{nabutovsky2006curvature}, this yields
\[
 A_{\min}(M,g)\ \le\ F(v,D)
\]
for some function $F$ depending only on $v$ and $D$. In that setting, however,
the dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made
explicit, since the constants coming from the Cheeger--Naber bubble--tree
decomposition \cite{cheeger2014regularity} were only known to exist abstractly.

In the present work we revisit this question in the more rigid Einstein setting.
For $(M^4,g)\in\Ee_1(4,v,D)$ the curvature solves an elliptic system, and
global Sobolev inequalities can be used, following Anderson
\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature
bounds and $\varepsilon$–regularity with constants depending only on a Sobolev
constant. This analytic input, combined with the Einstein specialization of the
Cheeger--Naber bubble--tree decomposition and an improved combinatorial filling
estimate, allows us to obtain an explicit upper bound for $A_{\min}(M,g)$ in
terms of $(v,D)$ and the Sobolev constants.

Our first main result is the following quantitative bound for minimal
$2$-varifolds in Einstein $4$-manifolds.

\begin{equation}\label{eq:Ein-assumptions}
 \Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad
 \diam(M,g)\le D,\quad H_1(M;\Z)=0.
\end{equation}

\begin{abstract}
We study the smallest area $A_{\min}(M,g)$ of a $2$-dimensional stationary integral varifold in a closed Einstein $4$-manifold $(M^4,g)$ with
\[
\Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad \diam(M,g)\le D,\quad H_1(M;\Z)=0.
\]
Building on the previous work \cite{LW_HF1} on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class there is an upper bound
\[
A_{\min}(M,g)\le F_{\Ein}(v,D),
\]
where $F_{\Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\varepsilon$–regularity constants for Einstein metrics.
\end{abstract}

In the present work we revisit this question in the more rigid Einstein setting.
For $(M^4,g)\in\Ee_1(4,v,D)$ the curvature solves an elliptic system, and
global Sobolev inequalities can be used, following Anderson
\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature
bounds and $\varepsilon$–regularity with constants depending only on a Sobolev
constant. This analytic input, combined with the Einstein specialization of the
Cheeger--Naber bubble--tree decomposition and an improved combinatorial filling
estimate, allows us to obtain an explicit upper bound for $A_{\min}(M,g)$ in
terms of $(v,D)$ and the Sobolev constants.

Our first main result is the following quantitative bound for minimal
$2$-varifolds in Einstein $4$-manifolds.

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} proceeds via a
homological filling inequality expressed in terms of the first homological
filling function $\HF_1$ of $(M,g)$.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property. If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

Theorem~\ref{thm:main} is then an immediate consequence of
Theorem~\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman
\cite{nabutovsky2006curvature}, which relates $A_{\min}(M,g)$ to $\HF_1$.
In the Einstein setting we can track all constants arising from the Sobolev
inequality, the $L^2$–curvature bound and the bubble–tree decomposition,
thus making $F_{\Ein}(v,D)$ completely explicit once the analytic data are fixed.

\begin{theorem}[Einstein $\varepsilon$-regularity via Sobolev]\label{thm:eps_reg_Einstein}
There exist dimensional constants $\varepsilon_{\mathrm{Ein}}>0$ and $c_{\mathrm{Ein}}>0$,
depending only on $C_S(v,D)$, such that the following holds.

\begin{theorem}[Nabutovsky--Rotman~{\cite{nabutovsky2006curvature}}]\label{thm:NR}
Let $M$ be a closed $n$-dimensional Riemannian manifold with $H_1(M;\Z)=0$
and diameter $\diam(M)=D$. Let $A_{\min}(M)$ denote the area of a smallest
$2$-dimensional stationary integral varifold in $M$. Then
\[
 A_{\min}(M)\;\le\;\frac{(n+1)!}{2}\,\HF_1(2D).
\]
In particular, for $n=4$,
\[
 A_{\min}(M)\;\le\;60\,\HF_1(2D).
\]
\end{theorem}

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

\begin{theorem}\label{thm:main}
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following
property.  If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a
smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
 A_{\min}(M,g)\ \le\ F_{\Ein}(v,D).
\]
Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev
constant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative
Einstein $\varepsilon$–regularity constants.
\end{theorem}

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} proceeds via a
homological filling inequality expressed in terms of the first homological
filling function $\HF_1$ of $(M,g)$.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\mathbb{Z})$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\mathbb{Z})$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

Theorem~\ref{thm:main} is then an immediate consequence of
Theorem~\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman
\cite{nabutovsky2006curvature}, which relates $A_{\min}(M,g)$ to $\HF_1$.
In the Einstein setting we can track all constants arising from the Sobolev
inequality, the $L^2$–curvature bound and the bubble–tree decomposition,
thus making $F_{\Ein}(v,D)$ completely explicit once the analytic data are fixed.

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} "proceeds via a homological filling inequality expressed in terms of the first homological filling function $\HF_1$ of $(M,g)$." One first proves Theorem~\ref{thm:Ein-fill}, i.e. a linear filling estimate for every singular Lipschitz $1$-cycle $C$ by a singular $2$-chain $E$ with \(\mass_2(E)\le f_1^{\Ein}(v,D)\,\mass_1(C)+f_2^{\Ein}(v,D)\), equivalently \(\HF_1(l)\le f_1^{\Ein}(v,D)\,l+f_2^{\Ein}(v,D)\). Then Theorem~\ref{thm:main} is "an immediate consequence" of Theorem~\ref{thm:Ein-fill} together with the min--max theorem of Nabutovsky--Rotman \cite{nabutovsky2006curvature}, "which relates $A_{\min}(M,g)$ to $\HF_1$." In the Einstein setting, the constants are tracked using "the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition," making $F_{\Ein}(v,D)$ explicit once the analytic data are fixed.

As in Larry Guth and David W. Rotman, \emph{The first homology of a Riemannian manifold and the first homological filling function}, the proof of the main theorem “proceeds via a homological filling inequality expressed in terms of the first homological filling function $\HF_1$ of $(M,g)$.” We first prove the following theorem.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

To prove the main theorem, we then combine this filling estimate with the min--max theorem of Nabutovsky--Rotman (Alexander Nabutovsky and Regina Rotman, \emph{Curvature-Free Bounds for the Least Area of a Surface}, 2006), which “relates $A_{\min}(M,g)$ to $\HF_1$.” In the Einstein setting, the constants are tracked using “the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,” making $F_{\Ein}(v,D)$ explicit once the analytic data are fixed.

\label{thm:main}
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following
property.  If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a
smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
 A_{\min}(M,g)\ \le\ F_{\Ein}(v,D).
\]
Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev
constant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative
Einstein $\varepsilon$–regularity constants.,

Question:
Let $v,D>0$. Suppose $(M^4,g)$ is a closed Einstein $4$-manifold in the class $\Ee_1(4,v,D)$, meaning that
\[
\Ric_g=\lambda g,\qquad |\lambda|\le 3,\qquad \Vol(M,g)\ge v,\qquad \diam(M,g)\le D,\qquad H_1(M;\mathbb Z)=0.
\]
Let $A_{\min}(M,g)$ denote the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$. Which conclusion about $A_{\min}(M,g)$ is valid under these hypotheses?

There exists a constant $F_{\Ein}(v,D)>0$, depending only on $v$ and $D$, such that
\[
A_{\min}(M,g)\le F_{\Ein}(v,D).
\]
Moreover, $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\varepsilon$-regularity constants.

Choices:
B.
There exists a universal constant $F_{\Ein}>0$, independent of $v$ and $D$, such that for every $(M^4,g)\in \Ee_1(4,v,D)$ one has
\[
A_{\min}(M,g)\le F_{\Ein}.
\]
Moreover, this bound can be written explicitly in terms of dimensional Einstein $\varepsilon$-regularity constants alone.
C.
There exists a constant $F_{\Ein}(v,D)>0$ such that
\[
A_{\min}(M,g)\le F_{\Ein}(v,D)
\]
for every $(M^4,g)\in \Ee_1(4,v,D)$.
D. For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property: if $(M^4,g)\in \Ee_1(4,v,D)$, then every $2$-dimensional stationary integral varifold in $(M,g)$ has area at most $F_{\Ein}(v,D)$. Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\varepsilon$-regularity constants.
E.
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property. If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
A_{\min}(M,g)\le F_{\Ein}(v,D),
\]
and one may choose $F_{\Ein}(v,D)$ to depend only on $v$, $D$, and the diameter term appearing in the Nabutovsky--Rotman estimate, without any use of the Sobolev constant, the global $L^2$-curvature bound, or Einstein $\varepsilon$-regularity constants.

dependence of the bound on v and D and analytic data

explicit description of how F_{\Ein}(v,D) is written in terms of Sobolev/L^2-curvature/\varepsilon-regularity data

smallest varifold versus arbitrary stationary varifold

necessity of Sobolev, L^2-curvature, and bubble-tree/\varepsilon-regularity input for explicit constants

The stem does not explicitly state the conclusion or uniquely identify choice A. It presents the hypothesis class and asks which uniform existence statement is valid, so there is no direct answer leakage.

The item is largely a theorem-recall question: the correct option is essentially the target theorem stated under the given hypotheses. However, the presence of nearby variants means it is not a pure verbatim restatement.

Some reasoning is needed to distinguish the strongest valid conclusion from weakened, overgeneralized, or tampered versions. Still, the task is mostly recognition/comparison of theorem formulations rather than genuine mathematical generation.

Several distractors are plausible and reflect realistic failure modes (omitting the H1 hypothesis, overstating independence of analytic constants, inserting an unjustified factor). But choice C is a weaker true statement, which creates ambiguity in a single-best-answer format and weakens distractor quality.

A reasonably constructed theorem-discrimination MCQ with no major answer leakage, but it is closer to recall than to generative reasoning, and the weaker-true distractor makes the single-correct-answer structure somewhat ambiguous.


Question:
Let $v,D>0$. Suppose $(M^4,g)$ is a closed Einstein $4$-manifold in the class $\Ee_1(4,v,D)$, meaning that
\[
\Ric_g=\lambda g,\qquad |\lambda|\le 3,\qquad \Vol(M,g)\ge v,\qquad \diam(M,g)\le D,\qquad H_1(M;\mathbb Z)=0.
\]
Let $A_{\min}(M,g)$ denote the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$. Which conclusion about $A_{\min}(M,g)$ is valid under these hypotheses?

Choices:
B.
There exists a universal constant $F_{\Ein}>0$, independent of $v$ and $D$, such that for every $(M^4,g)\in \Ee_1(4,v,D)$ one has
\[
A_{\min}(M,g)\le F_{\Ein}.
\]
Moreover, this bound can be written explicitly in terms of dimensional Einstein $\varepsilon$-regularity constants alone.
C.
There exists a constant $F_{\Ein}(v,D)>0$ such that
\[
A_{\min}(M,g)\le F_{\Ein}(v,D)
\]
for every $(M^4,g)\in \Ee_1(4,v,D)$.
D. For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property: if $(M^4,g)\in \Ee_1(4,v,D)$, then every $2$-dimensional stationary integral varifold in $(M,g)$ has area at most $F_{\Ein}(v,D)$. Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\varepsilon$-regularity constants.
E.
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property. If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
A_{\min}(M,g)\le F_{\Ein}(v,D),
\]
and one may choose $F_{\Ein}(v,D)$ to depend only on $v$, $D$, and the diameter term appearing in the Nabutovsky--Rotman estimate, without any use of the Sobolev constant, the global $L^2$-curvature bound, or Einstein $\varepsilon$-regularity constants.

The stem does not reveal the correct option explicitly or by an obvious cue. It only states the geometric hypotheses and asks for the valid conclusion.

The item is very close to a theorem-recall question: under the stated hypotheses, select the conclusion. It is not a pure verbatim restatement because the options vary in strength and in the dependence of the constant, but it remains largely theorem matching.

There is some reasoning pressure in distinguishing between a uniform bound, a bound for the minimum varifold versus all varifolds, and the role of explicit analytic inputs. However, the presence of a weaker true statement among the choices reduces the need for genuine generative reasoning.

Several distractors are mathematically plausible and target natural mistakes: over-uniformity, over-strong quantification, and incorrect dependence of constants. But choice C appears to be a weaker true consequence of A, so the distractor set is not cleanly single-correct.

Mathematically substantive and mostly free of answer leakage, but the item is weakened by theorem-like phrasing and, more seriously, by including a weaker statement that also appears valid, creating ambiguity in a single-answer MCQ.


The study of \hk \ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \cite[Thm.\ 1]{beauville}, they represent one of three ``fundamental" types of compact Kähler manifolds with \(c_1(K_X)=0\), together with abelian varieties and strict Calabi-Yau manifolds.
Their topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \Kah \ manifold, for example by the Fujiki relations, \cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \hkm s largely determines their geometry by the Torelli Theorem, see for example \cite{Markman_torelli}. On the other hand, the topology of \hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The aim of this note is to prove the following result:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\end{lemma}
\begin{proof}
Since \(H^3(-,\Z)_{\tors}\) is a birational invariant, by passing to a resolution of indeterminacies we may assume that \(f\) is everywhere defined. The result then follows from the projection formula.
\end{proof}
\begin{corollary}\label{cor:n-tors_of_Kum}
 The abelian group $H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}$ is $n$-torsion.
\end{corollary}
\begin{proof}
 The map \(K_{n-1}(A) \dashrightarrow \Kum_{n-1}(A)\) factors through the quotient by \(\mathfrak{S}_{n-1}\) acting on the first \(n-1\) entries of \(A^n\), yielding a map of degree \(n\).
 The quotient \(K_{n-1}(A)/\mathfrak{S}_{n-1}\) is birational to \(A^{[n-1]}\), hence we have a generically finite rational map of degree \(n\)
 \[A^{[n-1]} \dashrightarrow \Kum_{n-1}(A).\]
By \cite{Markman2007_integral} the group $H^3(A^{[n-1]},\Z)$ is torsion-free. The result then follows from Lemma \ref{lem:dn_tors} with \(N=1\).
\end{proof}
\section{Quotient by the alternating group}
Ideally, one would like to apply the argument of the previous section to the rational double cover $$A^{n-1} / \mathfrak A_n \dashrightarrow \Kum_{n-1}(A).$$ However, there is no smooth projective model \(Z\) of $A^{n-1} / \alt_n$ for which \(H^3(Z, \Z)_{\tors}\) is known.

Let $U \coloneqq K_{n-1}(A) \cap (A^n\setminus \Delta_2) \subset A^n$ and $V \coloneqq \Kum_{n-1}(A) \cap (A^{[n]} \setminus D_2) \subset A^{[n]}$.
For codimension reasons, avoiding the smaller diagonals does not change $H^3(-, \mathbb Z)_{\tors}$:
\begin{lemma}\label{lem:torsvkum}
 We have
 $H^3(V, \mathbb Z)_{\tors} \simeq H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}.$
\end{lemma}
\begin{proof}
We consider the stratification
\[D_{2}= Z_0 \supset Z_1\supset \cdots \]
where \(Z_{i+1}\) is the singular locus of \(Z_i\). By the long exact sequence of relative cohomology and the Thom isomorphism, we have
\[H^3(\Kum_{n-1}\setminus Z_{1},\Z)\simeq H^3(\Kum_{n-1}\setminus Z_{2},\Z) \simeq \dots \simeq H^3(\Kum_{n-1}(A), \mathbb Z),\]
see \cite[Lemma 11.13]{Voisin_book_I}. By the same argument we also have the exact sequence
$$ 0\to H^3(\Kum_{n-1} \setminus Z_1,\mathbb Z) \to H^3( \Kum_{n-1}(A) \setminus Z_0, \mathbb Z) \to \mathbb Z^{\pi_0(Z_0 \setminus Z_1)}.$$
Thus
\[H^3(\Kum_{n-1}(A),\Z)_{\tors}\simeq H^3(\Kum_{n-1}(A)\setminus Z_0,\Z)_{\tors}.\]
Since $V = \Kum_{n-1}(A) \setminus Z_0$, we conclude.
\end{proof}
Since the alternating group acts freely on $U$, the relation between the quotient $U / \alt_n$ and the open subset $V \subset \Kum_{n-1}(A)$ can be made precise as follows:
\begin{lemma}\label{lem:doublecover_V}
There exists a ramified double cover
 $$\pi \colon \Bl_{\overline{\Delta}} (U/\alt_n) \to V,$$
 where \(\overline{\Delta}\) is the image in \(U/\alt_n\) of \(\Delta\subset U\).
\end{lemma}
\begin{proof}
By \cite[Sec.\ 8, p.\ 770]{beauville}, we have
\[(\Bl_{\Delta} U )/ \mathfrak S_n \simeq V\]
Moreover, since $\mathfrak A_n$ acts freely on $U$ and stabilizes $\Delta$, we have
$$\Bl_{\Delta}(U) / \mathfrak A_n \simeq \Bl_{\overline{\Delta}}(U / \mathfrak A_n).$$
The action of \(\Z/2\Z\simeq\mathfrak{S}_n/\alt_n\) induced onto the quotient \(\Bl_{\Delta}(U) / \mathfrak A_n \) produces a double cover
 $$\Bl_{\overline\Delta}(U / \mathfrak A_n) \simeq (\Bl_{\Delta} U ) / \mathfrak A_n \to (\Bl_{\Delta} U) / \mathfrak S_n \simeq V,$$
 which is what we wanted.
 \end{proof}

\begin{proposition}\label{prop:spectral_sequence_argument}
 If there are integers $N_p$ for $1 \leq p \leq 3$ such that $$N_p \cdot H^p(\mathfrak A_n, H^{3-p}(K_{n-1}(A), \mathbb Z)) = 0,$$ then we have
 $$2N_1N_2N_3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
\end{proposition}
\begin{proof}
Since the action of $\mathfrak A_n$ on $U$ is free, there is the Cartan--Leray spectral sequence
$$H^p(\mathfrak A_n, H^q(U, \mathbb Z)) \Rightarrow H^{p+q}(U / \mathfrak A_n, \mathbb Z).$$
Note that for $p \leq 6$, by the same argument as in Lemma \ref{lem:torsvkum} we have $H^p(U, \mathbb Z) \simeq H^p(K_{n-1}(A), \mathbb Z)$, as the complement of $U \subset K_{n-1}(A)$ is of codimension four. In particular, $H^0(\mathfrak A_n, H^3(U, \mathbb Z)) \simeq H^3(K_{n-1}(A))^{\mathfrak A_n}$ is torsion-free.
It then follows that $H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}$ is an extension of subquotients of $H^{p}(\mathfrak A_n, H^{3-p}(K_{n-1}(A, \mathbb Z)))$ for $1 \leq p \leq 3$, which implies that \(N_1 N_2 N_3 \cdot H^3(U / \mathfrak A_n, \mathbb Z)_{\tors} = 0\) .
As blowing-up smooth subvarieties leaves $H^3(-, \mathbb Z)_{\tors}$ invariant, we have
$$H^3(\Bl_{\overline{\Delta}}(U / \mathfrak A_n), \mathbb Z)_{\tors} \simeq H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}.$$
By applying \cite[Thm.\ 5.4]{aguilarprieto} to the double cover of Lemma \ref{lem:doublecover_V} $$\pi \colon \Bl_{\overline\Delta}(U / \mathfrak A_n) \to V,$$
we have that \(\pi_* \pi^* \alpha =2 \alpha\) for any \(\alpha \in H^*(V,\Z)\), which allows us to conclude by Lemma \ref{lem:torsvkum}.
\end{proof}

\begin{theorem*}[Theorem \ref{thm:main}]
If \(n \neq 6\), then
 $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$
\end{theorem*}

\begin{proof}
On the one hand, we have $n \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ by Corollary \ref{cor:n-tors_of_Kum}. Note that this already implies the result for $n \in \{4, 6, 8\}$. On the other hand, by combining Proposition \ref{prop:spectral_sequence_argument} and Proposition \ref{prop:spectralval}, we have $2^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n = 3$ or $n \geq 8$ and $2 \cdot 12^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n=5,7$. The claim then follows by taking the greatest common divisor of $n$ and $2^3$ (respectively \ $2 \cdot 12^3$).
\end{proof}

\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}

For odd values of \(n\), this amounts to the following:
\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}
As already mentioned, the case \(n=2\) follows from \cite{kapfermenet}. This result confirms folklore expectations, as stated for example in \cite[Rem.\ 2.1.(ii)]{huyb2024}

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\begin{question}
    Does \(H^3(\Kum_{2n+1}(A),\Z)_{\tors}\) vanish for \(n \geq 1\) as well?
\end{question}

\subsection{Acknowledgements}
The authors would like to thank Daniel Huybrechts and Emanuele Macrì for their interest in the project, as well as Nick Addington for sharing his computations and suggesting that earlier bounds could be improved. The first author would like to thank Claire Voisin for the invitation to Paris, where this project was initiated.
Many thanks to Daniel Huybrechts and Paolo Stellari, respectively, for bringing up Question \ref{que:hk_torsion} at the ``K3 surfaces and friends"\footnote{Lorentz Center, Leiden, June 2025  (\href{https://www.lorentzcenter.nl/k3-surfaces-en-friends-brauer-groups-and-moduli.html}{\underline{link}}) }
and PRAGMATIC\footnote{University of Catania, Italy, September 2025 (\href{https://www.dmi.unict.it/pragmatic/docs/Pragmatic2025.html}{\underline{link}})} summer schools. We thank the organizers of said events for the wonderful research environment.
Both authors were supported by the ERC Synergy Grant 854361 HyperK.
The first author is grateful for the support provided by the International Max Planck Research School on Moduli Spaces at the Max Planck Institute for Mathematics in Bonn.

The strategy of the proof (inspired by and generalizing \,\cite{kapfermenet}) is as follows. To prove Theorem~\ref{thm:main}, one first considers “a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology,” which “forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion.” Then, “by considering a double cover of a suitable open of \(\Kum_{n-1}(A)\),” one shows that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) “is a \(2\)-group,” using “an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence.” When \(n\) is odd, combining these conclusions yields Corollary~\ref{cor:nice_statement_even}.

\label{thm:main}
If \(n \neq 6\), then
    $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$,

Question: Let \(A\) be an abelian surface, and for an integer \(n\ge 2\) let \(\Kum_{n-1}(A)\) denote the corresponding generalized Kummer variety. Write \(H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}\) for the torsion subgroup of its third integral cohomology. Which annihilation statement holds for this torsion subgroup?

If \(n\neq 6\), then \[\gcd(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\] If \(n=6\), then \[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]

Choices:
B.
For every integer \(n\ge 2\), one has
\[2^3\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
C.
If \(n\) is odd, then
\[H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
D.
If \(n\neq 6\), then
\[\operatorname{lcm}(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]
E.
If \(n\neq 6\), then
\[\gcd(2^2,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[2\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]

The stem gives only the setup and asks which conclusion holds; it does not state or strongly hint at the specific gcd(2^3,n) bound or the exceptional n=6 case.

This is essentially a theorem-selection item: the correct option is the precise statement of a known conclusion, though the presence of nearby variants prevents it from being a pure verbatim restatement.

The item requires distinguishing between closely related bounds (gcd vs lcm, sharp exponent, exceptional case), but success is driven more by recall/recognition of the exact theorem statement than by substantial generative reasoning.

The distractors are mathematically plausible and target realistic failure modes: overstrong uniform bounds, weaker true-but-not-best statements, replacing gcd by lcm, and tampering with the exceptional constant.

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


\label{thm:Cd-main}
For every $d \geq 1$ and every finite set $A\subset \mathbb{R}^m$ with exponential Freiman dimension $d$, we have
\begin{equation} \label{eq:main-tight}
  |A+A|\geq C_d(|A|-2^d)+3^d,
\end{equation}
where
\[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]

\begin{theorem}\label{thm:Cd-main}
For every $d \geq 1$ and every finite set $A\subset \R^m$ with exponential Freiman dimension $d$, we have
\begin{equation} \label{eq:main-tight}
  |A+A|\geq C_d(|A|-2^d)+3^d,
\end{equation}
where
\[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

A central theme in additive combinatorics suggests that in any abelian group, finite sets $A$ with small doubling
\[
 \sigma(A) \;:=\; \frac{|A+A|}{|A|}
\]
must exhibit strong additive structure. Here, $A+ A = \{ a+ a' : a,a' \in A\}$ denotes the sumset of $A$. Classical results of Freiman and their later refinements show that such sets are efficiently contained in
generalized arithmetic progressions or subspaces with bounded dimension; see, for example, Freiman's monograph
\cite{FreimanBook} and the book of Tao and Vu~\cite{TaoVu}.

When the abelian group above has no torsion, these questions are often related to studying the following problem: given a finite set $A \subset \mathbb{R}^m$, what type of geometric conditions must $A$ satisfy so as to ensure that the doubling $\sigma(A)$ grows rapidly, see \cite{Bilu, Chang}.
A basic instance of this is \emph{Freiman's lemma}, which states that if a finite set \(A\subset\mathbb{R}^m\) contains the vertices of a non-degenerate \(d\)-simplex, then
\[
 |A+A| \geq (d+1)|A| - \tfrac12 d(d+1) \geq d|A|/2.
\]
A non-degenerate \(d\)-simplex is a set $P \;=\; \{v_0,\,v_0+v_1,\dots,v_0+v_d\}$
with \(v_1,\dots,v_d\) linearly independent. So the mere presence of a simplex of dimension \(d\) forces the additive doubling $\sigma(A)$ to grow linearly in $d$.

This can be compared with the continuous setting where given a non-empty, compact set $\mathcal{A} \subset \mathbb{R}^d$, one can apply the Brunn--Minkowski inequality to deduce that $\mu(\mathcal{A} + \mathcal{A}) \geq 2^d \mu(\mathcal{A})$, with $\mu$ denoting the Lebesgue measure in $\mathbb{R}^d$. Thus it is natural to ask what local conditions can be prescribed for a finite set $A \subseteq \mathbb{R}^d$ to ensure that $\sigma(A)$ grows exponentially in $d$.

In this direction, Green and Tao \cite{GTcompressions} introduced the notion of \emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \subset \mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped
\[
 P \;=\; v_0 + \{0,1\} \cdot v_1 + \cdots + \{0,1\} \cdot v_d
\]
with \(v_1,\dots,v_d\) linearly independent. Green and Tao \cite{GTcompressions} proved that if $A \subset \mathbb{R}^m$ has exponential Freiman dimension $d$, then
\begin{equation} \label{sqrt2}
 |A+A| \geq (\sqrt{2})^{d} \,|A|.
 \end{equation}
Such inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\'{a}lv\"{o}lgyi--Zhelezov ~ \cite{PZ2020}. This circle of ideas has been significantly generalized in \cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\mathbb{R}$.

In \cite{GTcompressions}, Green and Tao noted that the inequality \eqref{sqrt2} is somewhat close to sharp: for example, if $A = \{0,1\}^d\subset\mathbb{R}^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that
\begin{equation}\label{eq:Cd-def}
 |A+A| \;\ge\; C_d\,|A| - O_d(1)
\end{equation}
holds for all finite sets $A \subset \mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that
$$(\sqrt{2})^{d} \leq C_{d} \leq 2(3/2)^{d-1},$$
with the upper bound following from considering the set $A = \{0,1\}^{d-1} \times \{0,1,\dots, N\}$.

Our main result closes the above exponential gap by determining the exact value of \(C_d\) for every $d \geq 1$.

\begin{equation} \label{sqrt2}
  |A+A| \geq (\sqrt{2})^{d} \,|A|.
  \end{equation}

In this direction, Green and Tao \cite{GTcompressions} introduced the notion of \emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \subset \mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped
\[
 P \;=\; v_0 + \{0,1\} \cdot v_1 + \cdots + \{0,1\} \cdot v_d
\]
with \(v_1,\dots,v_d\) linearly independent. Green and Tao \cite{GTcompressions} proved that if $A \subset \mathbb{R}^m$ has exponential Freiman dimension $d$, then
\begin{equation} \label{sqrt2}
 |A+A| \geq (\sqrt{2})^{d} \,|A|.
 \end{equation}
Such inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\'{a}lv\"{o}lgyi--Zhelezov ~ \cite{PZ2020}. This circle of ideas has been significantly generalized in \cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\mathbb{R}$.

In \cite{GTcompressions}, Green and Tao noted that the inequality \eqref{sqrt2} is somewhat close to sharp: for example, if $A = \{0,1\}^d\subset\R^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that
\begin{equation}\label{eq:Cd-def}
 |A+A| \;\ge\; C_d\,|A| - O_d(1)
\end{equation}
holds for all finite sets $A \subset \mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that
$$(\sqrt{2})^{d} \leq C_{d} \leq 2(3/2)^{d-1},$$
with the upper bound following from considering the set $A = \{0,1\}^{d-1} \times \{0,1,\dots, N\}$.

A perhaps surprising aspect of Theorem~\ref{thm:Cd-main} is that the natural upper bound example $A = \{0,1\}^{d-1} \times \{0,1,2,\dots, N\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\sigma(A)$.
Writing \([0,N] := \{0,1,\dots,N\}\)
for any \(N\in\N\), we describe these examples below.
\begin{align}
 A &= \{0,1\}^{d-1} \times [0,N], \label{eq:example-0.1}\\
 A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{d/2} \times [0,N]^{d/2}\bigr)
 &&\text{for \(d\) even}, \label{eq:example-0.2}\\
 A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{(d-1)/2} \times \{0,1\}\times [0,N]^{(d-1)/2}\bigr)
 &&\text{for \(d\) odd}. \label{eq:example-0.3}
\end{align}
Example~\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \{0,1,2\}^{\,d-1}\times[0,2N]$, so
$$|A+A| = 3^{\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$
Example~\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and
\[ A+ A = \{0,1,2\}^d \cup (\{0,1\}^{d/2} \times [0,N+1]^{d/2}) \cup (\{0\}^{d/2} \times [0, 2N]^{d/2}), \]
whence,
\[ \frac{|A+A|}{|A|} = \frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \rightarrow 2 \cdot 2^{d/2}-1 \]
as $N\rightarrow\infty$.
Finally, Example~\eqref{eq:example-0.3} satisfies $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and
\begin{align*}
 A+A &= \{0,1,2\}^d \cup (\{0,1\}^{(d-1)/2}\times \{0,1,2\}\times [0,N+1]^{(d-1)/2}) \\
 & \quad \cup (\{0\}^{(d-1)/2}\times \{0,1,2\}\times [0,2N]^{(d-1)/2}),
\end{align*}
and so, we get that
\begin{align*}
 \frac{|A+A|}{|A|} &= \frac{3\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\cdot (2N+1)^{(d-1)/2} - 3\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\
 &\rightarrow 3\cdot 2^{(d-1)/2} - \tfrac32
\end{align*}
as $N\rightarrow\infty$.

\begin{theorem} \label{lem:mainineq}
 Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
 \begin{equation} \label{anineq}
 \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
 \end{equation}
 where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
 \[
 C_d \;=\;
 \begin{cases}
 2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
 2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
 3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
 \end{cases}
\]
\end{theorem}

\begin{definition}[$i$-compression]
Let $A \subset \R^{d}$ be finite and fix $1 \le i \le d$. For each $\mathbf{u}\in\R^{d}$, set
\[
m(\mathbf{u}) := |A \cap H_{\mathbf{u}}|.
\]
We define the \emph{$i$-compression} $C_i(A)$ of $A$ by
\[
C_i(A)
 := \Bigl\{
(u_1,\dots,u_{i-1},t,u_{i+1},\dots,u_d) :
t\in \N_0, 0 \le t < m(\mathbf{u}) \text{ for some } \mathbf{u}
\Bigr\},
\]
that is, in each fiber $H_{\mathbf{u}}$, we replace $A \cap H_{\mathbf{u}}$ by the initial segment
$\{0,1,\dots,m(\mathbf{u})-1\}$.
\end{definition}

We begin by recording some useful notations. First, for vectors $u = (u_1, \dots, u_d)$ and $v = (v_1, \dots, v_d)$ in $\{0,1\}^d$, we define the Hamming distance $d(u,v)$ to be
\[ d(u,v) = \sum_{1 \leq i \leq d: \ u_i \neq v_i} 1 = \sum_{1 \leq i \leq d} |u_i - v_i|. \]
Moreover, we define
\[ |u| = d(u,0) = u_1 + \dots + u_d. \]
Next, recall that
\[C_d \;=\;
 \begin{cases}
 2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
 2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
 3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
 \end{cases}
\]

Just as before, we will redistribute some of the weight from $D_{k-m-2}$ to $D_{k-m-1}$ and $D_{k-m}$. By \nameref{lem:redistribution} twice, we have
\[X_{k-m-2} \geq \frac{k-m-1}{m+2} X_{k-m-1}\geq \frac{(k-m-1)(k-m)}{(m+2)(m+1)} X_{k-m}.\]
Thus we will be done if we can show that
\[\frac{(m+2)(m+1)}{(k-m-1)(k-m)}(C_d-D_{k-m})+\frac{m+1}{k-m}(C_d-D_{k-m-1})\leq D_{k-m-2} - C_d,\]
or equivalently,
\begin{equation} \label{eq:doddweight}
\frac{(m+2)(m+1)}{(k-m-1)(k-m)}\left(2^{k-m}-\frac{3}{2}\right)+\frac{m+1}{k-m}\left(2^{k-m-1}-\frac{3}{2}\right)\leq 3\cdot 2^{m-1}-2^{k-m-2} + \frac{3}{2}.
\end{equation}
Writing
\begin{align*}
 f_1(k,m) &= (m+2)(m+1)\left(2^{k-m}-\frac{3}{2}\right)+(m+1)(k-m-1)\left(2^{k-m-1}-\frac{3}{2}\right)\\
 &\quad - (k-m-1)(k-m)\left(3\cdot 2^{m-1}-2^{k-m-2}+\frac{3}{2}\right)
\end{align*}
for all $k,m \in \mathbb{Z}$, we see that \eqref{eq:doddweight} follows for $k,m$ if $f_1(k,m)\leq 0$.

The permutohedron $P_{d}$ lives inside the hyperplane $\left\{x \in \mathbb{R}^{d}:\ x_1+\ldots+x_d = \binom{d+1}{2}\right\}$, and has a number of remarkable properties (see for example \cite{Postnikov} or \cite{Stanley}).
For instance, when $d$ is even, the vertex set of $P_d$ contains $d/2$-dimensional affine cubes. Indeed, consider the $d/2$ disjoint adjacent transpositions
\[
t_j = (2j-1\;\;2j) \in S_{d}, \qquad j = 1,\dots,d/2.
\]
These transpositions commute and each of them has order two. Thereby, for each
$\varepsilon = (\varepsilon_1,\dots,\varepsilon_{d/2})\in\{0,1\}^{d/2}$, we can define the permutation
\[
\sigma_\varepsilon := t_1^{\varepsilon_1} \circ t_2^{\varepsilon_2} \cdots \circ t_{d/2}^{\varepsilon_{d/2}} \in S_{d}.
\]
Then the $2^{d/2}$ vertices of $P_{d}$ defined by
\[
\Bigl\{\,\bigl(\sigma_\varepsilon(1),\dots,\sigma_\varepsilon(d)\bigr)
 : \varepsilon\in\{0,1\}^{d/2} \Bigr\}
\]
form an affine copy of the ${d/2}$-dimensional cube $\{0,1\}^{d/2}$. Consequently, if a finite set \(A \subset \mathbb{Z}^d\) contains the vertex set of the permutohedron, then \(A\) automatically contains
an affine copy of \(\{0,1\}^{\lfloor d/2 \rfloor}\).
Our Theorem \ref{thm:Cd-main} therefore implies
\begin{equation} \label{cubeinP}
|A+A|\geq C_{d/2}(|A|-2^{d/2})+3^{d/2}.
\end{equation}

A perhaps surprising aspect of Theorem~\ref{thm:Cd-main} is  that the natural upper bound example $A = \{0,1\}^{d-1} \times \{0,1,2,\dots, N\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\sigma(A)$.
Writing \([0,N] := \{0,1,\dots,N\}\)
for any \(N\in\mathbb{N}\), we describe these examples below.
\begin{align}
  A &= \{0,1\}^{d-1} \times [0,N], \label{eq:example-0.1}\\
  A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{d/2} \times [0,N]^{d/2}\bigr)
      &&\text{for \(d\) even}, \label{eq:example-0.2}\\
  A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{(d-1)/2} \times \{0,1\}\times [0,N]^{(d-1)/2}\bigr)
      &&\text{for \(d\) odd}. \label{eq:example-0.3}
\end{align}
Example~\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \{0,1,2\}^{\,d-1}\times[0,2N]$, so
$$|A+A| = 3^{\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$
Example~\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and
\[ A+ A = \{0,1,2\}^d \cup (\{0,1\}^{d/2} \times [0,N+1]^{d/2}) \cup (\{0\}^{d/2} \times [0, 2N]^{d/2}),  \]
whence,
\[ \frac{|A+A|}{|A|} = \frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \rightarrow 2 \cdot 2^{d/2}-1 \]
as $N\rightarrow\infty$.
Finally, Example~\eqref{eq:example-0.3} satisfies  $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and
\begin{align*}
    A+A &= \{0,1,2\}^d \cup (\{0,1\}^{(d-1)/2}\times \{0,1,2\}\times [0,N+1]^{(d-1)/2}) \\
    & \quad \cup (\{0\}^{(d-1)/2}\times \{0,1,2\}\times [0,2N]^{(d-1)/2}),
\end{align*}
and so, we get that
\begin{align*}
    \frac{|A+A|}{|A|} &= \frac{3\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\cdot (2N+1)^{(d-1)/2} - 3\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\
    &\rightarrow 3\cdot 2^{(d-1)/2} - \tfrac32
\end{align*}
as $N\rightarrow\infty$.

Taking the minimum of these three upper bounds for each \(d\) recovers exactly
the values in Theorem~\ref{thm:Cd-main}. Furthermore, \eqref{eq:main-tight} is exactly tight for $d\leq 5$. Interestingly, for sufficiently large $d$, since $C_d<(3/2)^d$, Theorem \ref{thm:Cd-main} yields the bound
\[ |A+A|\geq C_d|A|+(3^d-2^dC_d), \]
wherein the error term is actually positive.

The main difficulty will be to show that $C_{d}$ is at least the indicated values above. Before we move on to the proof of Theorem \ref{thm:Cd-main} in \S2, let us start by discussing some of the high-level ideas behind it.

\medskip

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume
that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations
and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$, that is, for any \(u\in A\) and any $v \in \mathbb{Z}^d$ satisfying the inequality \(0\le v\le u\) coordinate-wise, we have \(v\in A\).  In particular, the
assumption \(\{0,1\}^d \subseteq A\) says that the ``bottom layer'' of \(A\) is a
complete discrete hypercube.  We then partition \(A\) according to which
coordinates lie below or above a fixed threshold, and obtain a parallel
partition of the sumset \(B := A+A\).  We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of \(B\)
in terms of the blocks of \(A\).

After a suitable renormalisation, this reduces Theorem~\ref{thm:Cd-main} to a
purely analytic inequality for families of non-negative real numbers
\(\{x_u\}_{u\in\{0,1\}^k}\) and \(\{y_w\}_{w\in\{0,1,2\}^k}\), indexed by slices
of the discrete (Hamming) hypercube and hypergrid, subject to two constraints:
a collection of inequalities of the form \(y_{u+v}\ge x_u+x_v\) arising from the sumset structure, and a monotonicity condition reflecting the downset
structure. The monotonicity condition is not strictly necessary, but it is convenient to have. In order to state this, we will require one further piece of notation, and so, given vectors $u=(u_1, \dots, u_d)$ and $v= (v_1, \dots, v_d)$ in $\mathbb{R}^d$, we say that $v\succeq u$ if $v_i \geq u_i$ for every $1 \leq i \leq d$.

\begin{theorem} \label{lem:mainineq}
    Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
    \begin{equation} \label{anineq}
        \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
    \end{equation}
    where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
    \[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

We remark that the constant $C_d$ in Theorem~\ref{thm:Cd-main} is tight for at least one value of $k$ for each $d$. In particular, for $d\leq 5$, tightness occurs at $k=d-1$, with $x_u=1$ for all $u$. For even $d\geq 6$, tightness occurs at $k=d/2$ with $x_0=1$ and $x_u=0$ for all $u\neq 0$. Finally, for odd $d\geq 7$, tightness occurs at $k=(d+1)/2$ with $x_0=x_{e_1}=1$ and $x_u=0$ for all other $u$. These equality cases naturally result in the constructions given in \eqref{eq:example-0.1}, \eqref{eq:example-0.2} and \eqref{eq:example-0.3}.

Theorem \ref{lem:mainineq} is proved by a
combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight.  The
dimension-dependent constants that arise along the way match precisely the
values of \(C_d\) asserted in Theorem~\ref{thm:Cd-main}.

\medskip

{\bf{Organization of the paper}}. In \S2 we discuss various prerequisites concerning compressions and prove how  Theorem~\ref{thm:Cd-main} follows from Theorem \ref{lem:mainineq}. We lay some preliminary brickwork for proving our main analytic inequality in \S3, which already turns out to be sufficient to deal with the small-dimensional cases \(d \in \{3,4,5\}\). The case when $d \geq 6$ is even is resolved in \S4, while the case when $d \geq 7$ is odd is analysed in \S5.  The odd case is slightly more involved compared to the even case, since the corresponding doubling constant $C_d$ is larger and the construction \eqref{eq:example-0.3} attaining ``equality'' is more complicated. In \S6, we will discuss a few natural future directions.

\medskip

{\bf{Acknowledgments.}} The authors would like to thank Ben Green for useful discussions. J.L. was supported by ERC Advanced Grant 883810. A.M. was supported by Leverhulme early career fellowship ECF-2025-148. C.P. was supported by NSF grant DMS-2246659. X.S. was supported by NSF grant DMS-2452462.

\medskip

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.

After a suitable renormalisation, this reduces Theorem~\ref{thm:Cd-main} to a purely analytic inequality for families of non-negative real numbers $\{x_u\}_{u\in\{0,1\}^k}$ and $\{y_w\}_{w\in\{0,1,2\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. This analytic inequality is stated as Theorem~\ref{lem:mainineq}.

Finally, Theorem~\ref{lem:mainineq} is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in Theorem~\ref{thm:Cd-main}.

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.

After a suitable renormalisation, to prove the main theorem we reduce to a purely analytic inequality for families of non-negative real numbers $\{x_u\}_{u\in\{0,1\}^k}$ and $\{y_w\}_{w\in\{0,1,2\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. We first prove the following theorem.

\begin{theorem} \label{lem:mainineq}
    Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
    \begin{equation} \label{anineq}
        \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
    \end{equation}
    where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
    \[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

Finally, the theorem above is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in the main theorem.

Question:
Let $d\ge 1$ be an integer, and let $A\subset \mathbb{R}^m$ be a finite set for some $m$. Write $A+A=\{a+a':a,a'\in A\}$. Say that $A$ has exponential Freiman dimension $d$ if $d$ is the largest positive integer for which $A$ contains the vertex set of a non-degenerate $d$-dimensional parallelepiped
\[
P=v_0+\{0,1\}\cdot v_1+\cdots+\{0,1\}\cdot v_d,
\]
where $v_1,\dots,v_d$ are linearly independent. Which statement holds for every such $d$ and every such finite set $A$?

One always has
\[
|A+A|\ge C_d\bigl(|A|-2^d\bigr)+3^d,
\]
where
\[
C_d=
\begin{cases}
2\bigl(\tfrac32\bigr)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6 \text{ even},\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd}.
\end{cases}
\]

Choices:
B.
One always has
\[
|A+A|\ge C_d\bigl(|A|-2^d\bigr)+3^d,
\]
where
\[
C_d=
\begin{cases}
2\bigl(\tfrac32\bigr)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6,\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd}.
\end{cases}
\]
C.
One always has
\[
|A+A|\ge C_d\bigl(|A|-2^d\bigr),
\]
where
\[
C_d=
\begin{cases}
2\bigl(\tfrac32\bigr)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6 \text{ even},\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd}.
\end{cases}
\]
D.
One always has
\[
|A+A|\ge C_d\,|A|-O_d(1),
\]
where
\[
C_d=
\begin{cases}
2\bigl(\tfrac32\bigr)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6 \text{ even},\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd},
\end{cases}
\]
and the implicit constant in \(O_d(1)\) may be chosen independently of \(d\) and \(m\).
E.
If \(A\subset \mathbb{R}^m\) contains the vertex set of some non-degenerate \(d\)-dimensional parallelepiped, then one always has
\[
|A+A|\ge C_d\bigl(|A|-2^d\bigr)+3^d,
\]
where
\[
C_d=
\begin{cases}
2\bigl(\tfrac32\bigr)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6 \text{ even},\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd}.
\end{cases}
\]

parity-dependent piecewise definition of C_d

largest-dimension hypothesis replaced by mere existence of a d-parallelepiped

The stem defines the setting and asks for a universal valid statement, but it does not reveal the bound, the constants, or the crucial quantifier structure of the correct answer.

This is essentially a theorem-identification item: the correct option is the exact theorem statement, while the others are nearby variants. It tests recall/recognition of the statement more than independent conclusion-drawing.

There is some reasoning pressure in distinguishing subtle changes in parity conditions, additive terms, quantifiers, and uniformity of constants, but the task is still mainly to recognize the exact theorem rather than generate it from mathematical reasoning.

The distractors are strong: they are close to the true statement and reflect realistic failure modes such as dropping the +3^d term, weakening the hypothesis, mishandling parity cases, or overstating uniformity.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is fairly tautological because it mostly asks for the exact statement rather than eliciting genuine generative reasoning.