New Bounds of Weyl Sums

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 , . . . , 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 to an infinite series of monotonically decreasing bounds, converging to our final result.

Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$

“…where 0 < α ≤ d. We extend the results of Chen and Shparlinski in [13] and Barron in [4] in T to the higher dimension T d for d ≥ 2, and obtain (1.14) for S α (N ) characterizes the extent of the occurrence for the square root cancellation for the Weyl sums ω N (x, t). It is easy to see that when (x, t) equals (0, 0) ∈ T d+1 or approximates (0, 0), constructive interference for ω N (x, t) occurs, that is…”

“…We recall the lower bound of S N,k (x k , t) given by Chen and Shparlinski in [13] and [15], where for…”

On generating functions in additive number theory, II: lower-order terms and applications to PDEs

“…Moreover, we have the trivial bound sup α,γ | f 1,k (α, α + γ ; Q)| Q, and it is known (see e.g. [6,Corollary 2.2]) that for independent variables γ , α we have | f 1,k (α, α + γ ; Q)| Q 1/2+ε almost everywhere. It turns out that in our case where only one of the variables is restricted to lie in the complement of a thin set while the other one ranges freely, the bound is appreciably larger.…”

“…Let Θ k denote the set of all θ ∈ R such that for almost all γ ∈ [0, 1) one has 12) and set θ k = inf Θ k . The size of θ k and related quantities has recently been studied by Chen and Shparlinski [6], building on work by Wooley [21]. Clearly, one sees that…”

On the cubic Weyl sum

“…More general restriction estimates for cubic sums were initiated by Bourgain [2] and further developed in [12,13,14]. See [5,6] for various bounds related to the metric behaviour of large values of Weyl sums and [3] for other metric results and an asymptotic formula for minor arc sums.…”