Exponential Sums and Riesz energies

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$andauniversal$c>0$ $$\sum _{i,j=1}^N\frac{X^2}{1+X^4\mathrm{\parallel }{x}_i-x_j{\mathrm{\parallel }}^4}\lesssim \sum _{\genfrac{}{}{0px}{}{k\in {\mathbb{Z}}^2}{\mathrm{\parallel }k\mathrm{\parallel }\le X}}{\mid \sum _{n=1}^N{e}^{2\pi i⟨k,x_n⟩}\mid }^2\lesssim \sum _{i,j=1}^N{X}^2{e}^{-c{X}^2\mathrm{\parallel }{x}_i-x_j{\mathrm{\parallel }}^2}\mathrm{.}$$ 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 $X\gtrsim N^{1\mathrm{/}2}$ both upper and lower bound match for maximally-separated point sets satisfying $\mathrm{\parallel }{x}_i-x_j\mathrm{\parallel }\gtrsim N^{-1\mathrm{/}2}$.