On the minimum size of restricted sumsets in cyclic groups

2013/05/09 by Bajnok, Béla
#11B25 #11P70 #20K01 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11B75 #Secondary: 05D99

paper · doi:10.48550/arxiv.1305.2141

Abstract

For positive integers n, m, and h, we let ρ (ℤ_n, m, h) denote the minimum size of the h-fold restricted sumset among all m-subsets of the cyclic group of order n. The value of ρ (ℤ_n, m, h) was conjectured for prime values of n and h=2 by Erdős and Heilbronn in the 1960s; Dias da Silva and Hamidoune proved the conjecture in 1994 and generalized it for an arbitrary h, but little is known about the case when n is composite. Here we exhibit an explicit upper bound for all n, m, and h; our bound is tight for all known cases (including all n, m, and h with n ≤ 40). We also provide counterexamples for conjectures made by Plagne and by Hamidoune, Lladó, and Serra.

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