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 ^α for some α > 0. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightlycurvedsequencewithsmallerror. Furthermore, weextendSzemerédi’stheorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence {⌊n ^a ⌋} _{n ∈A} contains arbitrarily long arithmetic progressions for every 1 ≤ a < 2 and every A ⊂ N with positive upper density. Using this corollary, we show that the set {⌊⌊p ^1/b ⌋ ^a ⌋ | 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 {⌊n ^a ⌋} ^∞ _n=1 does not contain any arithmetic progressions of length 3.