Small Fractional Parts of Polynomials and Mean Values of Exponential Sums

Abstract

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 $$\begin{align*} &\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},\end{align*}$$and derive upper bounds for fractional parts of polynomials in the shape $$\begin{align*} &\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),\end{align*}$$by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017).