# On Large Values of Weyl Sums

## TL;DR

This paper investigates the size and structure of exceptional coefficient sets where Weyl sums are unusually large, revealing they are quite large in terms of Baire category and Hausdorff dimension, especially for polynomials of degree d.

## Contribution

It demonstrates that the sets of coefficients with large Weyl sums are large in Baire category and Hausdorff dimension, and extends results to sums with a single monomial.

## Key findings

- Exceptional sets have positive Hausdorff dimension.
- Sets are large in Baire category and Hausdorff sense.
- Results apply to polynomials with poorly distributed fractional parts.

## Abstract

A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients $(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of $[0,1]^d$. We also use a different technique to give similar results for sums with just one monomial $xn^d$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.