On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$

S. I. Dimitrov; M. D. Lazarova

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arXiv · 2504.21333

On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$

S. I. Dimitrov·M. D. Lazarova

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Abstract

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $\frac{13}{14}<\gamma<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|\alpha p^2+\beta\|< p^{\frac{13-14\gamma}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1/\gamma}]$.

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S. I. Dimitrov, M. D. Lazarova. 2025-04-30. On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$. https://arxiv.org/abs/2504.21333

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