Strings of congruent primes in short intervals II

Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $ε$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes $p_r$, up to $X$, such that both $p_{r+1} - p_{r} < ε\log p_r$ and $p_{r} \equiv p_{r+1} \equiv a \bmod q$ simultaneously hold. As a lower bound for the number of primes satisfying the latter condition, the bound we obtain improves upon a bound obtained by D. Shiu.

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