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arXiv · 2004.02539

Metric theory of lower bounds on Weyl sums

Changhao Chen·Bryce Kerr·Igor Shparlinski
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Abstract

We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$, such that $$ \left|\sum_{n=1}^N \exp\left(2 \pi i\left(x_1n+\ldots+x_d n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d \ge 3$ and at least $3/2$ for $d=2$, where $c$ is a constant depending only on $d$. This improves the previous lower bound of the first and third authors for $d\ge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $xn^d$.

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Changhao Chen, Bryce Kerr, Igor Shparlinski. 2020-04-06. Metric theory of lower bounds on Weyl sums. https://arxiv.org/abs/2004.02539

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