Positive density for consecutive runs of sums of two squares

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 a₁ and a₂ mod q, and any r₁, r₂ 1, a positive density of sums of two squares begin a chain of r₁ consecutive sums of two squares, all of which are a₁ mod q, followed immediately by a chain of r₂ consecutive sums of two squares, all of which are a₂ 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 1, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.