Small fractional parts of polynomials and mean values of exponential sums
Abstract
Let ki\ (i=1,2,…,t) be natural numbers with k1>k2>·s>kt>0, k1≥ 2 and t<k1. Given real numbers αji\ (1≤ j≤ t,\ 1≤ i≤ s), we consider polynomials of the shape i(x)=α1ixk1+α2ixk2+·s+αtixkt, and derive upper bounds for fractional parts of polynomials in the shape 1(x1)+2(x2)+·s+s(xs), by applying novel mean value estimates related to Vinogradov's mean value theorem. Our results improve on earlier Theorems of Baker (2017).
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