  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 Szemer\'edi'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 $a\ge2$, the graph of
$\{\lfloor{n^a}\rfloor\}_{n=1}^\infty$ does not contain any arithmetic
progressions of length $3$.
