Small Fractional Parts of Polynomials and Mean Values of Exponential Sums.

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). [ABSTRACT FROM AUTHOR]