On small fractional parts of polynomial-like functions

Paolo Minelli

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arXiv · 2104.03232

On small fractional parts of polynomial-like functions

Paolo Minelli

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Abstract

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for $f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and any fixed $\xi\in [0,1]$ they obtained \[\min_{2\leq p\leq X} \Vert \xi \lfloor f(p)\rfloor \Vert\ll_{k,c,\epsilon} X^{-\rho_1(c,k)+\epsilon}\] for $\rho_1(c,k)>0$ explicitly given. In the present note, we improve upon their results in the case $c>k$ and $c>4$.

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Paolo Minelli. 2021-04-07. On small fractional parts of polynomial-like functions. https://arxiv.org/abs/2104.03232

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