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Exponential sums and Riesz energies. (English) Zbl 1423.11143

Summary: 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 \(\{x_1, \ldots, x_N \} \subset \mathbb{T}^2\), \(X \geq 1\) and a universal \(c > 0\) \[ \begin{aligned} \mathop{\sum}\limits_{i, j = 1}^N \frac{X^2}{1 + X^4 \| x_i - x_j \|^4} \lesssim \mathop{\sum}\limits_{\mathop{k \in \mathbb{Z}^2}\limits_{\| k \| \leq X}}\Big|\mathop{\sum}\limits_{n = 1}^N e^{2 \pi i \langle k, x_n \rangle}\Big|^2 \\ \lesssim \mathop{\sum}\limits_{i, j = 1}^N X^2 e^{- c X^2 \| x_i - x_j \|^2}. \end{aligned} \] 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 \(X \gtrsim N^{1 / 2}\) both upper and lower bound match for maximally-separated point sets satisfying \(\| x_i - x_j \| \gtrsim N^{- 1 / 2}\).

MSC:

11L07 Estimates on exponential sums
42B05 Fourier series and coefficients in several variables
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