Exponential sums and Riesz energies

JOURNAL OF NUMBER THEORY

卷:182

Article

Steinerberger, Stefan¹ 

[1] Yale Univ, Dept Math, 10 Hillhouse Ave, New Haven, CT 06511 USA

关键词: Riesz energy;    Discrepancy;    Exponential sums;    Fejer kernel;   

DOI  :  10.1016/j.jnt.2017.08.002

来源: Elsevier

【 摘 要 】

We bound an exponential sum that appears in the study of irregularities of distribution (the low-frequency Fourier energy of the sum of several Dirac measures) by geometric quantities: a special case is that for all {x(1),..., x(N)} subset of T-2; X >= 1 and a universal c > 0 Sigma (i,j,=1) (N) X-2/1+X-4 parallel to x(i) - x(j parallel to)4 less than or similar to Sigma(2)(kappa epsilon Z) parallel to kappa parallel to <= x vertical bar Sigma (N)(n=1) e(2 pi iota)vertical bar(2) less than or similar to Sigma (N)(i,j=1) X-e(2)-cX(2)parallel to x(i)-x(j)parallel to(2). since this exponential sum is intimately tied to rather subtle distribution properties of the points, we obtain nonlocal structural statements for near-minimizers of the Riesz-type energy. For X greater than or similar to N-1/2 both upper and lower bound match for maximally-separated point sets satisfying parallel to x(i)-x(j) parallel to greater than or similar to N-1/2. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free