Small fractional parts of polynomials and mean values of exponential sums

TL;DR

This paper establishes new upper bounds for the fractional parts of polynomial sums with multiple variables, using advanced mean value estimates related to Vinogradov's theorem, improving previous results by Baker.

Contribution

The paper introduces novel mean value estimates that enhance bounds on fractional parts of polynomial sums, advancing the understanding of exponential sums in number theory.

Findings

Improved bounds for fractional parts of polynomial sums.

Enhanced mean value estimates related to Vinogradov's theorem.

Refinement of earlier results by Baker (2017).

Abstract

Let $k_i\ (i=1,2,\ldots,t)$ be natural numbers with $k_1>k_2>\cdots>k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape $$\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},$$ and derive upper bounds for fractional parts of polynomials in the shape $$\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),$$ by applying novel mean value estimates related to Vinogradov's mean value theorem. Our results improve on earlier Theorems of Baker (2017).