A higher-dimensional Siegel-Walfisz theorem

The Green-Tao-Ziegler theorem provides asymptotics for the number of prime tuples of the form (ψ1(n),…,ψt(n)) when n ranges among the integer vectors of a convex body K⊂[−N,N]d and Ψ=(ψ1,…,ψt) is a system of affine-linear forms whose linear coefficients remain bounded (in terms of N). In the t=1 case, the Siegel-Walfisz theorem shows that the asymptotic still holds when the coefficients vary like a power of logN. We prove a higher-dimensional (i.e. t>1) version of this fact. We provide natural examples where our theorem goes beyond the one of Green and Tao, such as the count of arithmetic of progressions of step ⌊logN⌋ times a prime in the primes up to N. We also apply our theorem to the determination of asymptotics for the number of linear patterns in a dense subset of the primes, namely the primes p for which p−1 is squarefree. To the best of our knowledge, this is the first such result in dense subsets of primes save for congruence classes.