Small scale distribution of linear patterns of primes

Let $\mathrm{\Psi }$ 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 $\mathrm{\Psi }$: $$\sum _{x\in [-N,N{]}^d}\prod _{i=1}^t{\mathbf{1}}_{\mathcal{P}}({\psi }_i(x))\sim \frac{(2N{)}^d}{(\log N{)}^t}\prod _p{\beta }_p,$$ 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 $$\sum _{x\in B^{+}}\prod _{i=1}^t{\mathbf{1}}_{\mathcal{P}}({\psi }_i(x))>(1+{\delta }^{+})\frac{\mathrm{v}\mathrm{o}\mathrm{l}(B^{+})}{(\log N{)}^t}\prod _p{\beta }_p,$$ $$\sum _{x\in B^{-}}\prod _{i=1}^t{\mathbf{1}}_{\mathcal{P}}({\psi }_i(x))<(1-{\delta }^{-})\frac{\mathrm{v}\mathrm{o}\mathrm{l}(B^{-})}{(\log N{)}^t}\prod _p{\beta }_p\mathrm{.}$$