Positive density for consecutive runs of sums of two squares

TL;DR

This paper proves that for any odd squarefree modulus, there is a positive density of consecutive sums of two squares in specific residue classes, mirroring Maynard's prime sequence results.

Contribution

It establishes the existence of positive density chains of consecutive sums of two squares in arithmetic progressions, extending Maynard's prime sequence findings.

Findings

Positive density of chains of consecutive sums of two squares in arithmetic progressions.

Chains can be arbitrarily long for given residue classes.

Results extend analogies between sums of two squares and prime distributions.

Abstract

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus $q$, any two reduced congruence classes $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod $q$, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod $q$. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class $a$ mod $q$ and for any $r \ge 1$, a positive density of primes begin a sequence of $r$ consecutive primes, all of which are $a$ mod $q$.