Positive density for consecutive runs of sums of two squares

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 $q$.