Large Weyl sums and Hausdorff dimension

TL;DR

This paper determines the Hausdorff dimension of sets of coefficients for Gauss and Weyl sums that attain large values infinitely often, providing exact values and bounds for various polynomial degrees and sequences.

Contribution

It offers exact Hausdorff dimension values for Gauss sum coefficients and new upper bounds for Weyl sums with higher-degree polynomials, advancing understanding of large sum behaviors.

Findings

Exact Hausdorff dimension for Gauss sum coefficients when sums are large.

New upper bounds on Hausdorff dimension for Weyl sums with degree ≥ 3.

Matching known lower bounds for monomial sums near α close to 1.

Abstract

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $\alpha \in (1/2,1)$ achieve the order at least $N^{\alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $d\ge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $\alpha$ is close to $1$. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.