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arXiv · 2407.01611

Simultaneously Small Fractional Parts of Polynomials

Cheuk Fung Lau
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Abstract

Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier result of Maynard, who obtained the same exponent dependency on $k$ but not on $d$.

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Cheuk Fung Lau. 2024-06-27. Simultaneously Small Fractional Parts of Polynomials. https://arxiv.org/abs/2407.01611

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