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arXiv · 2210.03085

Small fractional parts of polynomials and mean values of exponential sums

Kiseok Yeon
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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 $α_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape $$φ_i(x)=α_{1i}x^{k_1}+α_{2i}x^{k_2}+\cdots+α_{ti}x^{k_t},$$ and derive upper bounds for fractional parts of polynomials in the shape $$φ_1(x_1)+φ_2(x_2)+\cdots+φ_s(x_s),$$ by applying novel mean value estimates related to Vinogradov's mean value theorem. Our results improve on earlier Theorems of Baker (2017).

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Kiseok Yeon. 2023-05-14. Small fractional parts of polynomials and mean values of exponential sums. https://arxiv.org/abs/2210.03085

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