Small‐scale distribution of linear patterns of primes

Let be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system : where are the corresponding local densities. In this paper, we demonstrate limits to equidistribution of these primes on small scales; we show the analog to Maier's result on primes in short intervals. In particular, we show that for all , there exist such that for sufficiently large, there exist boxes of sidelengths at least such that