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 $I_1=\{n_1-m,\dots ,n_1+m\}$ and $I_2=\{n_2-m,\dots ,n_2+m\}$ 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 $n\in I_1\cup I_2$.

35 pages. arXiv admin note: substantial text overlap with arXiv:2506.15641