Small values of Weyl sums

2019/07/06 by Chen, Changhao, Shparlinski, Igor E.
#11J83 #11K38 #11L15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.03101

Abstract

We prove that the set of (x₁, …, x_d)∈ [0,1)^d, such that \underlinelim_{N→ ∞}| ∑_n=1^Nexp(2 πi (x₁n+… + x_dn^d)) | =0, contains a dense G_δ set in [0,1)^d and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums ∑_n=1^Nexp(2πi x n^d), x ∈ [0,1).

Related

• Metric theory of lower bounds on Weyl sums
• Hausdorff dimension of the large values of Weyl sums
• Metric theory of Weyl sums
• Large Weyl sums and Hausdorff dimension
• Restricted mean value theorems and metric theory of restricted Weyl sums