On a conjecture of Brown concerning accessible sets

摘要

In this note we use a sequence constructed by Furstenberg in 1981 to disprove the following conjecture posed by Brown: If a set of positive numbers L is such that for any finite coloring of N there are arbitrarily long monochromatic sequences of distinct integers with all gaps in L, then for any finite coloring of N there are arbitrarily long monochromatic arithmetic progressions whose common differences belong to L. (c) 2004 Elsevier Inc. All rights reserved.