2008/03/18 by David J. Grynkiewicz, Grynkiewicz, David J.
Computer Science · #11B75 #11P70 #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Logic, Reasoning, and Knowledge #Number Theory (math.NT) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.0803.2601
openalex publication_date 2008/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let t≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let A\ppi B denote all the elements c with at least i representations of the form c=a+b, with a∈ A and b∈ B. For |A|, |B|≥ t, we show that either \be\Sumi=1t|A\ppi B|≥ t|A|+t|B|-2t2+1,\ee or else there exist A'⊆ A and B'⊆ B with \ber \nn l&:=&|A∖ A'|+|B∖ B'|≤ t-1, \nn A'\pptB'&=&A'+B'=A\pptB,and \nn \Sumi=1t|A\ppiB|&≥& t|A|+t|B|-(t-l)(|H|-ρ)-tl≥ t|A|+t|B|-t|H|,\eer where H is the (nontrivial) stabilizer of A\ppt B and ρ=|A'+H|-|A'|+|B'+H|-|B'|. In the case t=2, we improve (\refalmost) to |A\pp1B|+|A\pp2B|≥ 2|A|+2|B|-4.