247 research outputs found
Large restricted sumsets in general abelian group
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
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
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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