Small fractional parts of polynomials and mean values of exponential sums

Let $k_i(i=1,2,\dots ,t)$ be natural numbers with $k_1>k_2>\cdots >k_t>0$, $k_1\ge 2$ and $t<k_1\mathrm{.}$ Given real numbers ${\alpha }_{ji}(1\le j\le t,1\le i\le s)$, we consider polynomials of the shape $${\phi }_i(x)={\alpha }_{1i}{x}^{k_1}+{\alpha }_{2i}{x}^{k_2}+\cdots +{\alpha }_{ti}{x}^{k_t},$$ and derive upper bounds for fractional parts of polynomials in the shape $${\phi }_1(x_1)+{\phi }_2(x_2)+\cdots +{\phi }_s(x_s),$$ by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017).