247 research outputs found

    Large restricted sumsets in general abelian group

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    Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^S B).Comment: Paper submitted November 15, 2011. To appear European Journal of Combinatorics, special issue in memorian Yahya ould Hamidoune (2013

    Three-term arithmetic progressions and sumsets

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    Suppose that G is an abelian group and A is a finite subset of G containing no three-term arithmetic progressions. We show that |A+A| >> |A|(log |A|)^{1/3-\epsilon} for all \epsilon>0.Comment: 20 pp. Corrected typos. Updated references. Corrected proof of Theorem 5.1. Minor revisions

    On various restricted sumsets

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    For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.Comment: 11 pages; final version for J. Number Theor
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