On sums of two squares and a basis of order $2$

Abstract

Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$ and $I_{2}=\{n_2-m, \ldots, n_2+m\}$ such that $m = [(\log N) (\log \log N)^{1/325565}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and for any $n\in I_{1}\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\mathcal{R}$. In particular, we have $n(an+b)\notin \mathcal{R}$ for all $n\in I_{1}\cup I_{2}$.