Progression-free sets in finite abelian groups |
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Authors: | Vsevolod F Lev |
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Institution: | Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel |
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Abstract: | Let G be a finite abelian group. Write and denote by rk(2G) the rank of the group 2G.Extending a result of Meshulam, we prove the following. Suppose that A⊆G is free of “true” arithmetic progressions; that is, a1+a3=2a2 with a1,a2,a3∈A implies that a1=a3. Then |A|<2|G|/rk(2G). When G is of odd order this reduces to the original result of Meshulam.As a corollary, we generalize a result of Alon and show that if an integer k?2 and a real ε>0 are fixed, |2G| is large enough, and a subset A⊆G satisfies |A|?(1/k+ε)|G|, then there exists A0⊆A such that 1?|A0|?k and the elements of A0 add up to zero. When G is of odd order or cyclic this reduces to the original result of Alon. |
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