An improved lower bound related to the Furstenberg-Sárközy theorem

Abstract

Let D(n) denote the cardinality of the largest subset of the set {1, 2, . . . , n} such that the difference of no pair of elements is a square. A well-known theorem of Furstenberg and Sárközy states that D(n) = o(n). In the other direction, Ruzsa has proven that D(n) & nγ for γ = 1 2 ( 1 + log 7 log 65 ) ≈ 0.733077. We improve this to γ = 1 2 ( 1 + log 12… (More)

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