Prime Points in the Intersection of Two Sublattices

Let Ψ : Z → Z be an affine-linear map, and Ψ̇ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z) of Z. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture MN(s) for some finite s > 1, GreenTao [GT3] verified Dickson’s conjecture for Ψ of complexity s. Let Φ : Z → Z be an affine-linear map, and Φ̇ be its linear part. Suppose that Ψ(Z) ∩ Φ(Z) is nonempty and Ψ̇(Z)∩Φ̇(Z) is of finite index in Ψ̇(Z). In this paper, as an application of GreenTao’s theorem on Dickson’s conjecture, we study the distribution of prime points in the affine sublattice Ψ(Z) ∩ Φ(Z).