A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression (II)
Loading...
Date
2011-02
Authors
Liu, Yu-Ru
Spencer, Craig V.
Zhao, Xiaomei
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
Let G ≃ Z/k1Z ⊕ · · · ⊕ Z/kN Z be a finite abelian group with
ki
|ki−1 (2 ≤ i ≤ N). For a matrix Y = (ai,j) ∈ Z
R×S
satisfying
ai,1 + · · · + ai,S = 0 (1 ≤ i ≤ R), let DY (G) denote the maximal
cardinality of a set A ⊆ G for which the equations ai,1x1 + · · · +
ai,SxS = 0 (1 ≤ i ≤ R) are never satisfied simultaneously by
distinct elements x1, . . . , xS ∈ A. Under certain assumptions on
Y and G, we prove an upper bound of the form DY (G) ≤ |G|(C/N)
γ
for positive constants C and γ .
Description
This article is made available through Elsevier's Open Archive, https://doi.org/10.1016/j.ejc.2010.09.008. © 2010 Elsevier Ltd. All rights reserved.