Stretching convex domains to capture
many lattice points

This lattice point counting problem was originally motivated by a problem in high frequency shape optimization. Suppose that $\Omega\subset\mathbb{R}^{2}$ is an $a\times 1/a$ rectangular domain. Then the eigenvalues of the Dirichlet Laplacian $-\Delta_{\Omega}^{\mathcal{D}}$ are of the form

Thus, there is a bijection between the Dirichlet eigenvalues less than $\pi^{2}r^{2}$, and the positive lattice points in the ellipse $(x/a)^{2}+(ay)^{2}\leq r^{2}$. Hence, the statement $\lim_{r\rightarrow\infty}a(r)=1$ can also be interpreted as the statement that the rectangle that minimizes the Dirichlet eigenvalues in the high frequency limit approaches the square.

A dual result for eigenvalues of the Neumann Laplacian $-\Delta_{\Omega}^{\mathcal{N}}$ was established by van den Berg, Bucur & Gittins [17], where as before $\Omega\subset\mathbb{R}^{2}$ is an $a\times 1/a$ rectangular domain. The eigenvalues of $-\Delta_{\Omega}^{\mathcal{N}}$ are of the form

and hence, the Neumann eigenvalues less than $\pi^{2}r^{2}$ are in bijection with the nonnegative lattice points in the ellipse $(x/a)^{2}+(ay)^{2}\leq r^{2}$. In terms of a lattice point problem, van den Berg, Bucur & Gittins proved that if

then $\lim_{r\rightarrow\infty}a(r)=1$. That is, the ellipse which captures the least nonnegative lattice points in the large area limit approaches the circle; equivalently, the rectangle which maximizes the Neumann eigenvalues in the high frequency limit approaches the square.

Subsequently, the result for Dirichlet eigenvalues was generalized to three dimensions by Gittins & van den Berg [18], and recently to $d$-dimensions for both the Dirichlet and Neumann cases by Gittins & Larson [5]. Specifically, Gittins & Larson show that in $\mathbb{R}^{d}$ the cuboid of unit measure which minimizes the Dirichlet eigenvalues in the high frequency limit approaches the cube, and similarly, that the cuboid of unit measure which maximizes the Neumann Laplacian eigenvalues in the high frequency limit approaches the cube. Both the Dirichlet and Neumann cases have corresponding lattice point problems analogous to the $2$-dimensional case. The Dirichlet case corresponds to the following lattice point problem. Suppose $A=\operatorname{diag}(a_{1},a_{2},\ldots,a_{d})$ is a positive diagonal matrix of determinant $1$. For $r\geq 1$, let

Then the result of [5] implies that $\lim_{r\rightarrow\infty}\|A-\operatorname{Id}\|_{\infty}=0$. Moreover, Gittins & Larson [5] show the following error estimate holds for dimensions $d\geq 2$:

and furthermore, they show a slightly improved error rate holds for $d>5$ by applying a result of Götze [6]. In this paper, we establish a similar rate of convergence for general convex domains. We note that the Neumann case in $d$-dimensions has an analogous dual formulation to the $2$-dimensional case, and similar error estimates were established in [5]. We remark that results concerning the optimization of eigenvalues of the Laplacian with perimeter constraints have also been considered [2, 4, 16]. In particular, Bucur & Freitas [4] proved that any sequence of minimizers of the Dirichlet eigenvalues with a perimeter constraint converges to the unit disk in the high frequency limit, and that among $k$-sided polygons, any sequence of minimizers converges to the regular $k$-sided polygon in the high frequency limit.

Laugesen & Liu [10] and Ariturk & Laugesen [3] generalized the two dimensional case of the above lattice point counting problems to certain classes of convex and concave curves. In particular, their results imply that among $p$-ellipses $\left|x/a\right|^{p}+\left|ay\right|^{p}=r^{p}$ for $p\in(0,\infty)\setminus\{1\}$ the $p$-ball captures the most positive lattice points. More generally, Laugesen & Liu and Ariturk & Laugesen show that the curves which capture the most positive integer lattice points in the large area limit are balanced: the distance from the origin to their points of intersection with the $x$-axis and $y$-axis are equal.

The results of [3, 10] exclude the $p=1$ case. In contrast to other values of $p$, for the $p=1$ case (where $p$-ellipses are right triangles) Ariturk & Laugesen and Laugesen & Liu conjectured that the optimal triangle

“does not approach a 45-45-90 degree triangle as $r\rightarrow\infty$. Instead one seems to get an infinite limit set of optimal triangles” (from [10]).

The author and Steinerberger [11] recently proved this conjecture and showed that the limit set is fractal of Minkowski dimension at most $3/4$. Furthermore, all triangles which are optimal for infinitely many infinitely large areas have rational slopes, are contained in $[1/3,3]$, and have $1$ as a unique accumulation point.

Suppose that $R\subset\mathbb{R}^{d}$ is a cuboid, and let $N_{R}(\lambda)$ denote the number of eigenvalues of the Dirichlet Laplacian $-\Delta_{R}^{\mathcal{D}}$ that are less than $\lambda$

Then $N_{R}(\lambda)$ has a two-term asymptotic formula in terms of the volume $|R|$ and surface area $|\partial R|$ of $R$

where $v_{d}$ is the volume of the unit ball in $\mathbb{R}^{d}$. A similar asymptotic formula holds for the eigenvalues of the Neumann Laplacian $-\Delta_{R}^{\mathcal{N}}$, see [8]. We remark that in 1913, Weyl, see page 199 of [21], speculated about the existence of such a two-term asymptotic formula for more general domains in $\mathbb{R}^{3}$, and the problem of establishing the above two-term asymptotic formula for general domains became known as Weyl’s Conjecture, see [8]. In 1980, Ivrii [9] proved that this two-term asymptotic formula indeed holds for more general domains under certain conditions. Suppose that $R$ is an $a_{1}\times a_{2}\times\cdots\times a_{d}$ cuboid. Then the eigenvalues of the Dirichlet Laplacian $-\Delta^{\mathcal{D}}_{R}$ are of the form

and thus, the eigenvalues of $-\Delta_{R}^{\mathcal{D}}$ less than $\pi^{2}r^{2}$ are in bijection with the positive lattice points in the ellipsoid $(x_{1}/a_{1})^{2}+(x_{2}/a_{2})^{2}+\cdots+(x_{d}/a_{d})^{2}\leq r^{2}$. If this cuboid $R$ has unit measure $|R|=1$, then its surface area $|\partial R|$ is given by

Substituting $\lambda=\pi^{2}r^{2}$ into the above two-term asymptotic formula for $N_{R}(\lambda)$ for this cuboid gives

We emphasize that the error term in this asymptotic formula depends implicitly on the cuboid $R$, and therefore, this asymptotic formula by itself is insufficient to determine which cuboid $R$ of unit measure maximizes $N_{R}(\lambda)$ as $\lambda\rightarrow\infty$. Addressing this issue is the main challenge in [1, 5, 17, 18]. If we were to ignore the error term, then maximizing $N_{R}(\lambda)$ among cuboids of unit measure would be equivalent to minimizing $\operatorname{tr}A^{-1}$. By the arithmetic mean geometric mean inequality

with equality if and only if $A=\operatorname{Id}$. Since $2\operatorname{tr}A^{-1}=|\partial R|$, this inequality can be interpreted as an isoperimetric inequality for cuboids: the surface area of a cuboid of unit measure is greater than or equal to the surface area of the unit cube with equality if and only if the cuboid is the unit cube. Ultimately, after the main challenge of dealing with the error term has been appropriately handled, this isoperimetric inequality for cuboids is the reason that the cube is asymptotically optimal in [1, 5, 17, 18]. In this paper, we consider a positive lattice point counting problem in more general domains; specifically, we study the number of positive lattice points in a fixed convex domain $\Omega$ that has been scaled by $r\geq 1$ and stretched by a linear transformation $A$ represented by a positive diagonal matrix of determinant $1$. In this generalized setting, the term $\operatorname{tr}A^{-1}$ similarly arises, and we use Fourier analysis to develop lattice point counting results which lead to uniform error estimates for optimal stretching problems.

In the following we compute an asymptotic expansion for $\#\{n\in\mathbb{Z}^{d}\cap A(r\Omega)\}$ which includes both the effects of the diagonal transformation $A$ and scaling factor $r$; we use this expansion to derive uniform error estimates for an optimal stretching problem. The resulting theorem extends the results of [1, 3, 5, 17, 18]. We note that we do not completely recover the results of [3, 10], since our approach can only represent curves which can be realized as the boundary of a convex domain whose boundary has nowhere vanishing Gauss curvature. However, there is some hope that the presented framework could be used to fully generalize the results of Laugesen & Liu [10] and Ariturk & Laugesen [3] by using more delicate bounds on the decay of the Fourier transform. Thus the presented approach may be useful for proving further generalizations.

Suppose that $\Omega\subset\mathbb{R}^{d}$ is a bounded convex domain whose boundary $\partial\Omega$ is $C^{d+2}$ and has nowhere vanishing Gauss curvature. Furthermore, suppose that $\Omega$ is symmetric with respect to each coordinate hyperplane and balanced in the following sense.

In the following remark, we describe specifically how the implicit constants in Theorem 2.1 depend on $\Omega\subset\mathbb{R}^{d}$, and provide conditions under which the expansion and convergence rate results in Theorem 2.1 hold uniformly over a family of domains.

When $\Omega\subset\mathbb{R}^{d}$ is a $d$-dimensional ellipsoid, the Theorem implies the Dirichlet Laplacian results of Gittins & Larson [5]. Applying the Theorem for dimension $d=2$ recovers the original result of Antunes & Freitas [1] and agrees with the error estimate in [10]. Specifically, in dimension $d=2$, the set of all positive diagonal matrices of determinant $1$ is the $1$-parameter family $A=\operatorname{diag}(1/a,a)$ for $a>0$ and the result of the Theorem can be stated as:

As a second Corollary we can establish dual results for the nonnegative lattice point problem, which corresponds to the Neumann Laplacian results of Gittins & Larson [5].

The assumption $1\leq\|A^{-1}\|_{\infty}\leq Cr$ is needed for the expansion to hold, but not necessary for the convergence results as it serves to avoid the case where $A(r\Omega)$ contains more than order $r^{d}$ lattice points on the coordinate hyperplanes which is clearly non-optimal for the $\operatorname*{argmin}$. The proof of this Corollary follows from the arguments in Step 3.3 of the proof of Theorem 2.1.

In the following we describe a generalization of Theorem 2.1. In particular, we remark how Theorem 2.1 can be generalized to domains $\Omega$ which are not necessarily symmetric with respect to each coordinate hyperplane by considering the lattice points $\{n\in\mathbb{Z}^{d}:n_{j}\not=0,\forall j=1,\ldots,d\}$ rather than the positive lattice points $\mathbb{Z}^{d}_{>0}$.

Before discussing the technical details, we describe the proof strategy. The proof is divided into five steps. First, we establish a new lattice point counting result which holds uniformly over a family of positive diagonal matrices $A$ with determinant $1$. Specifically, we show that

when $1\leq\|A^{-1}\|_{\infty}\leq Cr$ where $C>0$ is a fixed constant. The key observation in the proof of this expansion is that the classical Fourier analysis techniques used to study the Gauss circle problem can be applied if the indicator function of the domain $A(r\Omega)$ is mollified by a bump function which has been appropriately stretched and scaled. Second, as a consequence of this result we show that the number of lattice points on the coordinate hyperplanes is, as expected, equal to $\left(\operatorname{tr}A^{-1}\right)r^{d-1}$ plus an appropriate error term; specifically,

where $\Omega_{j}$ denotes the intersection of $\Omega$ with the hyperplane orthogonal to the $j$-th coordinate vector. Third, we combine these two results to produce a positive lattice point counting result for $A(r\Omega)$:

Fourth, we show that this positive lattice point counting result implies that

where the $\operatorname*{argmax}$ is taken over positive diagonal matrices $A$ of determinant $1$. We note that clearly $\|A(r)^{-1}\|_{\infty}=\mathcal{O}(r)$ since otherwise $A(r\Omega)$ will not contain any positive lattice points; we proceed by considering two cases. First, we consider the case where

where our positive lattice point counting result for $A(r\Omega)$ provides no information because the error term is order $r^{d}$. However, in such an extreme situation we are able to independently show non-optimality. Second, we assume that

and use the positive lattice point counting result to show convergence. Specifically, we assume

Substituting $\|A^{-1}\|_{\infty}=\psi(r)r$ into the positive lattice point counting result for $A(r\Omega)$ gives

where $C$ is a fixed constant. Since $\psi\rightarrow 0$ the term in the parentheses we eventually be positive, and hence, comparison to the case where $A=\operatorname{Id}$ implies convergence. Fifth, and finally, given the fact that $A(r)$ converges to $\operatorname{Id}$ we establish the convergence rate

using a standard arithmetic mean geometric mean argument.

The proof of the main result relies on three lemmata: two geometric in nature, and one related to the decay of the Fourier transform of indicator functions of convex domains with nowhere vanishing Gauss curvature. Lemmata 3.1 and 3.2 are proved in this section, and are motivated by similar technical results in [14], while Lemma 3.3 appears in standard references, cf., [13, 14, 15].

The application of this Lemma to lattice point counting problems occurs when developing bounds for indicator functions in terms of mollified indicator functions. Specifically, Lemma 3.1 is typically applied as follows. Suppose $\chi$ is the indicator function for the set $\Omega\subset\mathbb{R}^{d}$, and define

which is the indicator function for $r\Omega$. Let $\varphi$ denote a $C^{\infty}$ bump function supported on the unit ball which integrates to $1$. Set

Suppose $c>0$ is chosen in accordance to Lemma 3.1. Then we claim that for all $x\in\mathbb{R}^{d}$

where $*$ denotes convolution. Indeed, by the choice of $c>0$, for all $|y|\leq\delta$

Therefore, if $\chi_{r}(x)=1$, then

In the proof of the Theorem in the following section a similar result is required. However, in this case the indicator functions and bump functions under consideration are transformed by a positive diagonal linear transformation $A$ of determinant $1$. More precisely, we consider

where $A$ is a positive diagonal matrix of determinant $1$. Let

In the following Lemma, we repeat the above analysis to develop upper and lower bounds for $\chi_{A,r}$ in terms of the smoothed version $\chi_{A,r,\delta}$.

That is to say, the Fourier transform of an indicator function decays one order better than the Fourier transform of the corresponding surface carried measure $d\mu$, which decays like

The proof of this lemma involves an integration by parts of an expression for the Fourier transform of the corresponding surface carried measure, which is where the extra order of convergence arises.

For clarity, we have divided the proof of the Theorem into five steps. First, we fix notation. We say

for a fixed constant $C_{h}>0$ only depending on $h$. Throughout the proof, we assume that $r\geq 1$, $d\geq 2$, and that $\Omega\subset\mathbb{R}^{d}$ is a bounded convex domain with a $C^{d+2}$ boundary $\partial\Omega$ with nowhere vanishing Gauss curvature. In particular, we assume that

Let $A=\operatorname{diag}(a_{1},a_{2},\ldots,a_{d})$ denote a positive diagonal matrix of determinant $1$. Without loss of generality we may assume that $a_{1}\leq a_{2}\leq\ldots\leq a_{d}$, and set $a=1/a_{1}=\|A^{-1}\|_{\infty}$.