The ternary Goldbach–Vinogradov theorem with almost equal primes from the Beatty sequence

Abstract

Let α 1, α 2, α 3, β 1, β 2, β 3 be real numbers with α 1, α 2, α 3 >1. Suppose that each individual α i is of a finite type and that at least one pair $\alpha_{i}^{-1}$ , $\alpha_{j}^{-1}$ is also of a finite type. In this paper we prove that every large odd integer n can be represented as $$n=p_{1}+p_{2}+p_{3}, $$ with p i =n/3+O(n 2/3(logn) c ) and $p_{i}\in\mathcal{B}_{i}$ , where c>0 is an absolute constant and $\mathcal{B}_{i}$ denotes the so-called Beatty sequence, i.e. $$\mathcal{B}_{i}=\bigl\{n\in\mathbb{N}: n=[\alpha_{i}m+ \beta_{i}] \mbox { for some } m\in\mathbb{Z}\bigr\}. $$