格点和特征值的最优拉伸
Optimal stretching for lattice points and eigenvalues
摘要：
我们的目标是最大化的数量的第一象限格点在一个凸区域相对于相互拉伸的坐标方向。最优域被证明是渐近平衡的，这意味着当“半径”接近无穷大时，拉伸因子接近1。特别地，该结果意味着在所有p-椭圆（或Lamé曲线）中，当半径接近无穷大时，p-圆包围最多的第一象限格点。p=2的情况下对应于矩形上的Dirichlet Laplacian的高特征值的最小化，因此我们的工作推广了Antunes和Freitas的结果。同样，我们推广了范登伯格，Bucur和Gittins的Neumann特征值最大化结果。p=1的情况仍然是开放的：当面积趋于无穷大时，第一象限中的哪个直角三角形的两条边都沿着轴，将包围最多的晶格点？
英文摘要：
We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result implies that among all p-ellipses (or Lamé curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity. The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. The case p=1 remains open: 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?
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