Strings of special primes in arithmetic progressions

The Green–Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu’s Theorem to certain subsets of the primes such as primes of the form ⌊ π n ⌋ and some of arithmetic density zero such as primes of the form ⌊ n log log n ⌋ .