On the distribution of αp² modulo one in the intersection of two Piatetski--Shapiro sets

2023/12/05 by Chu, Junyi, Jinjiang Li, Li, Xiaotian +3
Mathematics · #Analytic Number Theory Research #Mathematical Dynamics and Fractals #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2312.02775

Abstract

Let \lfloor t\rfloor denote the integer part of t∈ℝ and ‖x‖ the distance from x to the nearest integer. Suppose that 1/2<γ₂<γ₁<1 are two fixed constants. In this paper, it is proved that, whenever α is an irrational number and β is any real number, there exist infinitely many prime numbers p in the intersection of two Piatetski--Shapiro sets, i.e., p=\lfloor n₁^1/γ₁\rfloor=\lfloor n₂^1/γ₂\rfloor, such that \beginequation* ‖αp²+β‖

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