This paper proves that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. More precisely, 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 $\alpha>0$. Furthermore, we extend Szemer\'edi's theorem to a theorem for slightly curved sequences. As a corollary, it follows that the graph of the sequence of the integer parts of $\{n^{a}\}_{n\in A}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and every $A\subset\mathbb{N}$ with positive upper density. We also prove that the same graph does not contain any arithmetic progressions of length $3$ for every $a\ge2$. 查看全文>>