Comparison estimates for linear forms in additive number theory

Abstract

Abstract. Let R be a commutative ring R with 1 R and with group of units R × . Let Φ = Φ(t 1 , . . . , t h ) = h i=1 ϕ i t i be an h-ary linear form with nonzero coefficients ϕ 1 , . . . , ϕ h ∈ R. Let M be an R-module. For every subset A of M , the image of A under Φ isFor every subset I of {1, 2, . . . , h}, there is the subset sum s I = i∈I ϕ i . Let S(Φ) = {s I : ∅ = I ⊆ {1, 2, . . . , h}}. The difference set A − A is the image of A under the linear form Υ(t 1 , t 2 ) = t 1 − t 2 and the h-fold sumset hA is the image of A under the linear form Φ(t 1 , t 2 , . . . , t h ) = t 1 + t 2 + · · · + t h . Equivalently, Ruzsa constructed a subset A of the Z-module M = Z/mZ such thatThis is a significant result in additive number theory. In this paper, we extend Ruzsa's theorem to a large class of pairs of linear forms Υ and Φ. Let R be a commutative ring with multiplicative identity 1 R = 0. We denote the group of units in R by R × . Associated to every sequence (ϕ 1 , . . . , ϕ h ) of nonzero