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

TL;DR

The paper advances the understanding of how quadratic phases $\alpha p^2$ distribute modulo $1$ for primes $p$ restricted to the Piatetski-Shapiro sequence $p=[n^{1/\gamma}]$ with $\frac{13}{14}<\gamma<1$. Using a Vaughan-type framework with a $1$-periodic majorant and a carefully designed sum $\Gamma$, the authors reduce the problem to obtaining a nontrivial lower bound for a smoothed prime sum and upper bounds for auxiliary oscillatory sums. They establish concrete exponential-sum bounds and apply Dirichlet approximation and smoothing techniques to prove that infinitely many Piatetski-Shapiro primes satisfy $\|\alpha p^2+\beta\|<p^{\frac{13-14\gamma}{29}+\varepsilon}$, under the stated conditions. The results illustrate how hybrid approaches merging Vinogradov-type estimates with thin prime sets can yield new distributional statements, potentially extendable with sharper exponential-sum techniques.

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 $α$ is irrational and $β$ is real then for any fixed $\frac{13}{14}<γ<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp^2+β\|< p^{\frac{13-14γ}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1/γ}]$.

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5