Exponential Sums and Riesz energies

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 $|x_i -x_j| \gtrsim N^{-1/2}$.