Small‐scale distribution of linear patterns of primes.

Let Ψ=(ψ1,⋯,ψt):'Zd→Rt$\Psi =(\psi _1,\hdots, \psi _t):'\mathbb {Z}^d\rightarrow \mathbb {R}^t$ 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 Ψ$\Psi$: ∑x∈[−N,N]d∏i=1t1P(ψi(x))∼(2N)d(logN)t∏pβp,$$\begin{equation*} \sum _{x\in [-N,N]^d} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) \sim \frac{(2N)^d}{(\log N)^t} \prod _{p} \beta _p, \end{equation*}$$where βp$\beta _p$ 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 λ>t/d$\lambda > t/d$, there exist δλ±>0$\delta _\lambda ^\pm > 0$ such that for N$N$ sufficiently large, there exist boxes B±⊂[−N,N]d$B^\pm \subset [-N, N]^d$ of sidelengths at least (logN)λ$(\log N)^\lambda$ such that ∑x∈B+∏i=1t1P(ψi(x))>(1+δλ+)vol(B+)(logN)t∏pβp,$$\begin{equation*} \sum _{x\in B^+} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) > (1+\delta _{\lambda }^+) \frac{\textrm {vol}(B^+)}{(\log N)^t} \prod _{p}\beta _p, \end{equation*}$$∑x∈B−∏i=1t1P(ψi(x))<(1−δλ−)vol(B−)(logN)t∏pβp.$$\begin{equation*} \sum _{x\in B^-} \prod _{i=1}^t \mathbb {1}_{\mathcal {P}}(\psi _i(x)) < (1-\delta _{\lambda }^-) \frac{\textrm {vol}(B^-)}{(\log N)^t} \prod _{p}\beta _p. \end{equation*}$$ [ABSTRACT FROM AUTHOR]