On large values of Weyl sums

Abstract

A special case of the Menshov–Rademacher theorem implies for almost all polynomials x1Z+…+xdZd∈R[Z] of degree d for the Weyl sums satisfy the upper bound|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩽N1/2+o(1),N→∞. Here we investigate the exceptional sets of coefficients (x1,…,xd) 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.