Small values of Weyl sums

Changhao Chen; Igor E. Shparlinski

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arXiv · 1907.03101

Small values of Weyl sums

Changhao Chen·Igor E. Shparlinski

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Abstract

We prove that the set of $(x_1, \ldots, x_d)\in [0,1)^d$, such that $$ \underline{\lim}_{N\to \infty}\left| \sum_{n=1}^N\exp(2 \pi i (x_1n+\ldots + x_dn^d)) \right| =0, $$ contains a dense $\mathcal{G}_\delta$ set in $[0,1)^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$ \sum_{n=1}^N\exp(2\pi i x n^d), \qquad x \in [0,1). $$

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Changhao Chen, Igor E. Shparlinski. 2019-07-06. Small values of Weyl sums. https://arxiv.org/abs/1907.03101

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