Mean square discrepancy bounds for the number of lattice points in large convex bodies
英文摘要:
We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.
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Alexander Iosevich, Eric Sawyer, Andreas Seeger
We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.
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We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that $(1-\varepsilon)K\subset P\subset K$.
Let B denote a three-dimensional body of rotation, with respect to one coordinate axis, whose boundary is sufficiently smooth and of bounded nonzero Gaussian curvature throughout, except for the two boundary points on the axis of rotation, where the curvature may vanish. For a large real variable t, we are interested in the number A(t) of integer points in the linearly dilated body tB, in particular in the lattice discrepancy P(t) = A(t) - volume(tB). We are able to evaluate the contribution of the boundary points of curvature zero to P(t), with a remainder that is fairly small in mean-square.
Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study
针对环面上有限点集相对于平移和缩放的非光滑边界凸体的L^2平均差异,在傅里叶变换衰减的各向异性假设下,证明了匹配的上下界,得到速率N^{1 - (d+1)/(d^2+d-1)},并恢复d=2时的已知指数2/5。
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