Small scale distribution of linear patterns of primes

Abstract

Let $\Psi$ 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 $\Psi$: $$\sum_{x\in [-N,N]^d} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(\psi_i(x)) \sim \frac{(2N)^d}{(\log N)^t} \prod_{p} \beta_p,$$ where $\beta_p$ 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 $\lambda > 1$, there exist $\delta_\lambda^\pm > 0$ such that for $N$ sufficiently large, there exist boxes $B^\pm\subset [-N, N]^d$ of sidelengths at least $(\log N)^\lambda$ such that $$\sum_{x\in B^+} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(\psi_i(x)) > (1+\delta^+) \frac{\mathrm{vol}(B^+)}{(\log N)^t} \prod_{p}\beta_p,$$ $$\sum_{x\in B^-} \prod_{i=1}^t \mathbf{1}_{\mathcal{P}}(\psi_i(x)) < (1-\delta^-) \frac{\mathrm{vol}(B^-)}{(\log N)^t} \prod_{p}\beta_p.$$