On Snevily's conjecture and restricted sumsets
Zhi-Wei Sun
Abstract
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(n-1)/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<j≤ n. 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.
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