On the Minimum Size of Signed Sumsets in Elementary Abelian Groups

2014/12/04 by Bajnok, Bela, Matzke, Ryan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1412.1609

Abstract

For a finite abelian group G and positive integers m and h, we let ρ(G, m, h) = min \|hA| : A ⊆ G, |A|=m\ and ρ_± (G, m, h) = min \|h_± A| : A ⊆ G, |A|=m\, where hA and h_± A denote the h-fold sumset and the h-fold signed sumset of A, respectively. The study of ρ(G, m, h) has a 200-year-old history and is now known for all G, m, and h. In previous work we provided an upper bound for ρ_± (G, m, h) that we believe is exact, and proved that ρ_± (G, m, h) agrees with ρ(G, m, h) when G is cyclic. Here we study ρ_± (G, m, h) for elementary abelian groups G; in particular, we determine all values of m for which ρ_± (ℤ_p², m, 2) equals ρ(ℤ_p², m, 2) for a given prime p.

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• The Minimum Size of Signed Sumsets
• Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers
• Sumsets with a minimum number of distinct terms
• A generalization of sumsets modulo a prime
• Direct and Inverse Problems for Restricted Signed Sumsets -- I