拉伸凸区域以捕获多个格点
Stretching convex domains to capture many lattice points
摘要：
我们考虑了$\mathbb{R}^d$中严格凸区域的最优拉伸问题，该区域关于每个坐标超平面对称，其中拉伸是指通过行列式$1$的对角矩阵的变换.具体来说，我们证明了拉伸凸区域捕获最积极的晶格点在大体积极限是平衡的：$(d-1)$维措施的交叉点的域与每个坐标超平面是平等的。我们的结果扩展了Antunes & Freitas，van den Berg，Bucur & Gittins，Ariturk & Laugesen，van den Berg & Gittins，and Gittins & Larson的结果.该方法的动机是由傅里叶分析技术用来证明经典的$\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =πr^2 + \mathcal{O}(r^{2/3})$结果的高斯圆问题。
英文摘要：
We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the $(d-1)$-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes & Freitas, van den Berg, Bucur & Gittins, Ariturk & Laugesen, van den Berg & Gittins, and Gittins & Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =πr^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.
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