Metric theory of lower bounds on Weyl sums

Abstract

We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1{)}^d$, such that $$\left|\sum _{n=1}^N\exp \left(2\pi i\left(x_{1}n+\dots +x_d{n}^d\right)\right)\right|\ge c{N}^{1/2}$$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d\ge 3$ and at least $3/2$ for $d=2$, where $c$ is a constant depending only on $d$. This improves the previous lower bound of the first and third authors for $d\ge 3$. We also obtain similar bounds for the Hausdorff dimension of the set of large sums with monomials $x{n}^d$.

All results of this preprint are now included, together with several other results, in 2011.09306 - "Metric theory of Weyl sums", by C. Chen, B. Kerr, J.Maynard, I. Shparlinski