Sublattices of finite index

Abstract

Assuming the Gowers Inverse conjecture and the M\"obius conjecture for the finite parameter s, Green-Tao verified Dickson's conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate Green-Tao's theorem on Dickson's conjecture, and prove that, if L is the range of a linear map of complexity s, and L1 is a sublattice of L of finite index, then L1 is the range of a linear map of complexity s.

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