On Snevily's conjecture and restricted sumsets

• Authors: Zhi-Wei Sun
• Venue: Journal of Combinatorial Theory Series A
• Year: 2003

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)(n/2) 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. This extends a recent result of Dasgupta, Károlyi, Serra and Szegedy on Snevily's conjecture. Actually stronger results on sumsets with polynomial restrictions are obtained in this paper.