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 for some . 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 Szemerédi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence contains arbitrarily long arithmetic progressions for every and every with positive upper density. Using this corollary, we show that the set contains arbitrarily long arithmetic progressions for every and b>1. We also prove that, for every , the graph of does not contain any arithmetic progressions of length 3.