A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression (II)

Authors:
Yu-Ru Liu;Craig V. Spencer;Xiaomei Zhao
Affiliations:
Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1;Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506, United States;Department of Mathematics, Huazhong Normal University, Wuhan, Hubei, 430079, China
Venue:
European Journal of Combinatorics
Year:
2011

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On subsets of Abelian groups with no 3-term arithmetic progression

Journal of Combinatorial Theory Series A
On subsets of finite Abelian groups with no 3-term arithmetic progressions

Journal of Combinatorial Theory Series A
A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Designs, Codes and Cryptography

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Abstract

Let G~Z/k"1Z@?...@?Z/k"NZ be a finite abelian group with k"i|k"i"-"1(2@?i@?N). For a matrix Y=(a"i","j)@?Z^R^x^S satisfying a"i","1+...+a"i","S=0(1@?i@?R), let D"Y(G) denote the maximal cardinality of a set A@?G for which the equations a"i","1x"1+...+a"i","Sx"S=0(1@?i@?R) are never satisfied simultaneously by distinct elements x"1,...,x"S@?A. Under certain assumptions on Y and G, we prove an upper bound of the form D"Y(G)@?|G|(C/N)^@c for positive constants C and @c.