The h-critical number of finite abelian groups

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

paper · pdf · doi:10.48550/arxiv.1412.4058

arxiv created 2014/12/12 · arxiv updated 2014/12/15

Abstract

For a finite abelian group G and a positive integer h, the unrestricted (resp.~restricted) h-critical number χ(G,h) (resp.~χ (G,h)) of G is defined to be the minimum value of m, if exists, for which the h-fold unrestricted (resp.~restricted) sumset of every m-subset of G equals G itself. Here we determine χ(G,h) for all G and h; and prove several results for χ (G,h), including the cases of any G and h = 2, any G and large h, and any h for the cyclic group ℤ_n of even order. We also provide a lower bound for χ (ℤ_n,3) that we believe is exact for every n---this conjecture is a generalization of the one made by Gallardo, Grekos, et al.~that was proved (for large n) by Lev.

Related

• More on the h-critical numbers of finite abelian groups
• Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers
• The critical number of finite abelian groups
• The Minimum Size of Signed Sumsets
• On the Minimum Size of Signed Sumsets in Elementary Abelian Groups