# Hausdorff dimension of the large values of Weyl sums

## TL;DR

This paper investigates the Hausdorff dimension of sets of vectors with large Weyl sums, providing an upper bound that complements previous lower bounds, thus advancing understanding of their fractal structure.

## Contribution

It introduces an upper bound for the Hausdorff dimension of vectors with large Weyl sums, complementing earlier lower bounds and refining the fractal analysis of these sets.

## Key findings

- Established an upper bound for the Hausdorff dimension of large Weyl sum sets.
- Complemented previous lower bounds, narrowing the possible dimension range.
- Enhanced understanding of the fractal geometry of Weyl sum exceptional sets.

## Abstract

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| \ge N^{\alpha} $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.