On the distribution of $\alpha p^2$ modulo one with prime $p$ of a special form

TL;DR

This paper proves that infinitely many primes of a specific almost-prime form satisfy a Diophantine approximation condition involving quadratic polynomials, improving previous results in the distribution of such primes.

Contribution

It establishes the existence of infinitely many primes of the form p+2 = an almost-prime with at most 4 prime factors that approximate a linear form with quadratic terms, for a wider range of parameters.

Findings

Infinitely many primes p with p+2 = -prime satisfy  p^2 +  approximations.

Improves previous bounds on the approximation exponent  for primes of special form.

Extends understanding of the distribution of primes in quadratic Diophantine problems.

Abstract

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there exist infinitely many primes $p$, such that $\|\alpha p^2+\beta\|<p^{-\theta}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.