Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772479 | Journal of Number Theory | 2018 | 25 Pages |
Abstract
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 {x1,â¦,xN}âT2, Xâ¥1 and a universal c>0âi,j=1NX21+X4âxiâxjâ4â²âkâZ2âkââ¤X|ân=1Ne2Ïiãk,xnã|2â²âi,j=1NX2eâcX2âxiâxjâ2. 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 Xâ³N1/2 both upper and lower bound match for maximally-separated point sets satisfying âxiâxjââ³Nâ1/2.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Stefan Steinerberger,