Distribution of solutions to systems of congruences in balls

2024/03/08 by Harm, Michael
#11D79 #11K38 #11T23 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.05078

Abstract

Let G₁,…, G_n∈ \mathbbF_p[X₁,…,X_m] be n polynomials in m variables over the finite field \mathbbF_p of p elements. For any sufficiently large prime p and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of G=(G₁,…, G_n), we study various properties regarding the distribution of the vectors by fractional parts (\ (G₁(x))/(p)\,⋯,\ (G_n(x))/(p)\)∈ \mathbbTⁿ,\hspace10pt x∈ \mathbbF_p^m. We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.

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