凸曲线下格点的最优拉伸
Optimal stretching for lattice points under convex curves
摘要：
假设我们计算在第一象限中具有相等截距的凸递减曲线下方的正整数格点。然后在坐标方向上拉伸，以保持曲线下的区域，并再次计算晶格点。选择哪种拉伸因子将使晶格点数最大化？我们表明，最佳的拉伸因子接近$1$的面积接近无穷大。特别地，当$0<p<1$在具有$s>0$的$p$ -椭圆$|sx|^p+|s^{-1}y|^p=r^p$中时，包围最第一象限格点的一个接近$p$ -圆（$s=1$）作为$r \to \infty$。Antunes和Freitas建立了$p=2$案例，并由Laugesen和Liu推广为$1<p<\infty$。案例$p=1$仍然是开放的，其中的问题是：在第一象限中，哪些直角三角形的两条边沿着轴，将包围最多的晶格点，因为面积趋于无穷大？我们的结果$p<1$借给支持的猜想，在所有的维度，矩形盒的体积，最小化$n$ -th特征值的Dirichlet拉普拉斯算子将接近一个立方体作为$n \to \infty$。这个猜想在四维和更高的维度上仍然是开放的。
英文摘要：
Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count lattice points. Which choice of stretch factor will maximize the lattice point count? We show the optimal stretch factor approaches $1$ as the area approaches infinity. In particular, when $0<p<1$, among $p$-ellipses $|sx|^p+|s^{-1}y|^p=r^p$ with $s>0$, the one enclosing the most first-quadrant lattice points approaches a $p$-circle ($s=1$) as $r \to \infty$. The case $p=2$ was established by Antunes and Freitas, with generalization to $1<p<\infty$ by Laugesen and Liu. The case $p=1$ remains open, where the question is: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity? Our results for $p<1$ lend support to the conjecture that in all dimensions, the rectangular box of given volume that minimizes the $n$-th eigenvalue of the Dirichlet Laplacian will approach a cube as $n \to \infty$. This conjecture remains open in dimensions four and higher.
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