On Snevilyʹs conjecture and restricted sumsets

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A1,…,An (n>1) be finite subsets of G with cardinality k>0, and let b1,…,bn be pairwise distinct elements of G with odd order. We show that for every positive integer m⩽(k−1)/(n−1) there are more than (k−1)n−(m+1)(n2) sets {a1,…,an} such that a1∈A1,…,an∈An, and both ai≠aj and mai+bi≠maj+bj (or both mai≠maj and ai+bi≠aj+bj) for all 1⩽i<j⩽n. This extends a recent result of Dasgupta, Károlyi, Serra and Szegedy on Snevilyʹs conjecture. Actually stronger results on sumsets with polynomial restrictions are obtained in this paper.