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

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
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.

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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