Comparison estimates for linear forms in additive number theory

TheoremLetϒ(t1,…,tg)=∑i=1gυitiandΦ(t1,…,th)=∑i=1hφitibe linear forms with nonzero coefficients in the ring R. If{0,1}⊆S(ϒ)andS(Φ)⊆R×, then for everyε>0andc>1there exist a finite R-module M with|M|>cand a subset A of M such thatϒ(A∪{0})=Mand|Φ(A)|<ε|M|.