By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in \mathbb{N}$, $\alpha \in (d,d+1)$, and integers $k\ge d+2$ and $r\ge 1$, there exist infinitely many $n\in \mathbb{N}$ such that the sequence ${(\lfloor {(n+rj)}^{\alpha }\rfloor )}_{j=0}^{k-1}$ is represented as $\lfloor {(n+rj)}^{\alpha }\rfloor =p(j)$, $j=0,1,\dots ,k-1$, by using some polynomial $p(x)\in \mathbb{Q}[x]$ of degree at most d. In particular, the above sequence is an arithmetic progression when $d=1$. In this paper, we show the asymptotic density of such numbers n as above. When $d=1$, the asymptotic density is equal to $1/(k-1)$. Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields.

通过使用Frantzikinakis和Wierdl的作品，我们可以看到所有 $d\in \mathbb{ñ}$， $\alpha \in （d，d+\mathrm{1个}）$和整数 $ķ\ge d+2$ 和 $\mathrm{[R}\ge \mathrm{1个}$，无限地存在 $ñ\in \mathbb{ñ}$ 这样的顺序 ${（\lfloor {（ñ+\mathrm{[R}Ĵ）}^{\alpha }\rfloor ）}_{Ĵ=0}^{ķ-\mathrm{1个}}$ 表示为 $\lfloor {（ñ+\mathrm{[R}Ĵ）}^{\alpha }\rfloor =p（Ĵ）$， $Ĵ=0，\mathrm{1个}，\dots ，ķ-\mathrm{1个}$，通过使用多项式 $p（X）\in \mathbb{问}[X]$最多度d。特别地，以上序列是算术级数，当$d=\mathrm{1个}$。在本文中，我们显示了上述数字n的渐近密度。什么时候$d=\mathrm{1个}$，渐近密度等于 $\mathrm{1个}/（ķ-\mathrm{1个}）$。尽管在以上结果中任意确定了公共差r，但我们也研究了r不固定的情况。通过使用属于Hardy字段的函数可以概括本文的大多数结果。