# Small values of Weyl sums

## TL;DR

This paper investigates the behavior of Weyl sums and generalized Gaussian sums, showing that the set of points where these sums tend to zero is dense, contains a large subset, and has positive Hausdorff dimension.

## Contribution

It establishes that the set of points with vanishing Weyl sums is dense, large in Hausdorff dimension, and extends results to generalized Gaussian sums.

## Key findings

- The set of points with Weyl sums tending to zero is dense in [0,1)^d.
- This set has a positive Hausdorff dimension.
- Similar properties are proven for generalized Gaussian sums.

## 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 $$ \sum_{n=1}^N\exp(2\pi i x n^d), \qquad x \in [0,1). $$