# New bounds of Weyl sums

## TL;DR

This paper improves bounds on Weyl sums and discrepancy of fractional parts of polynomials by extending Wooley's method with new ideas, including a self-improving approach and generalizations to projections, also providing Hausdorff dimension bounds.

## Contribution

It introduces novel bounds for Weyl sums, extends results to projections of coefficients, and develops a self-improving method for tighter bounds.

## Key findings

- Enhanced metric bounds on Weyl sums
- New bounds on discrepancy of polynomial fractional parts
- Upper bounds on Hausdorff dimension of large Weyl sum sets

## Abstract

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a ``self-improving'' approach, which leads an infinite series of monotonically decreasing bound, converging to our final result.