Arithmetic Progressions in the Graphs of Slightly Curved Sequences

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to 1/x(alpha) for some alpha > 0. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. Furthermore, we extend Szemeredi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence {Left perpendicularn(alpha)Right perpendicular }(n is an element of A) contains arbitrarily long arithmetic progressions for every 1 <= a < 2 and every A subset of N with positive upper density. Using this corollary, we show that the set {Left perpendicular Left perpendicular p(1/b) (alpha)Right perpendicular(a)Right perpendicular vertical bar p prime } contains arbitrarily long arithmetic progressions for every 1 <= a < 2 and b > 1. We also prove that, for every a >= 2, the graph of { Left perpendicularn(alpha)Right perpendicular }(n)(infinity)(=1) does not contain any arithmetic progressions of length 3.