On Large Values of Weyl Sums (1901.01551v5)
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 $xnd$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.