Arithmetic Progressions in the Graphs of Slightly Curved Sequences

Abstract

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 Szemerédi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence $\{\lfloor n^a\rfloor {\}}_{n\in A}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and every $A\subset \mathbb{N}$ with positive upper density. Using this corollary, we show that the set $\{ \lfloor{\lfloor{p^{1/b}}\rfloor^a}\rfloor \mid \text{$p$ prime} \}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and $b>1$. We also prove that, for every $a\ge 2$, the graph of $\{\lfloor n^a\rfloor {\}}_{n=1}^{\infty }$ does not contain any arithmetic progressions of length $3$.

19 pages; revised Section 1 and the proof of Theorem A.4 and added Section 2