Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.1

Arithmetic Progressions in the Graphs of Slightly Curved Sequences


Kota Saito and Yuuya Yoshida
Graduate School of Mathematics
Nagoya University
Furo-cho, Chikusa-ku, Nagoya, 464-8602
Japan

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 $\{ \lfloor{\lfloor{p^{1/b}}\rfloor^a}\rfloor
\vert \text{$p$\space prime} \}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and b>1. We also prove that, for every $a\ge2$, the graph of $\{\lfloor{n^a}\rfloor\}_{n=1}^\infty$ does not contain any arithmetic progressions of length 3.


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Received October 22 2018; revised versions received February 21 2019; February 22 2019. Published in Journal of Integer Sequences, February 22 2019.


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