Exponential Sums and Riesz energies

TL;DR

This paper establishes bounds for exponential sums related to irregularities of distribution on the torus, linking them to geometric configurations and deriving structural properties of near-minimizers of Riesz energies.

Contribution

It introduces geometric bounds for exponential sums on the torus and connects these bounds to the distribution properties of point sets, providing new insights into Riesz energy minimizers.

Findings

Bounds for exponential sums in terms of geometric quantities.

Structural properties of near-minimizers of Riesz energies.

Matching bounds for maximally-separated point sets when X is large.

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 $\left\{ x_1, \dots, x_N\right\} \subset \mathbb{T}^2$, $X \geq 1$ and a universal $c>0$ $$ \sum_{i,j=1}^{N}{ \frac{X^2}{1 + X^4 \|x_i -x_j\|^4}} \lesssim \sum_{k \in \mathbb{Z}^2 \atop \|k\| \leq X}{ \left| \sum_{n=1}^{N}{ e^{2 \pi i \left\langle k, x_n \right\rangle}}\right|^2} \lesssim \sum_{i,j=1}^{N}{ X^2 e^{-c X^2\|x_i -x_j\|^2}}.$$ 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. In the regime $X \gtrsim N^{1/2}$ both upper and lower bound match for maximally-separated point sets satisfying $\|x_i -x_j\| \gtrsim N^{-1/2}$.