论文标题
在Piatetski-Shapiro序列中由多项式表示的有限序列的分布
Distributions of Finite Sequences Represented by Polynomials in Piatetski-Shapiro Sequences
论文作者
论文摘要
通过使用Frantzikinakis和wierdl的工作，我们可以看到，对于所有$ d \ in \ mathbb {n} $，$α\ in（d，d+1）$，以及integers $ k \ ge d+2 $ d+2 $ and $ r \ ge1 $ $（\ lfloor {（n+rj）^α} \ rfloor）_ {j = 0}^{k-1} $表示为$ \ lfloor {（n+rj）^α} \ rfloor = p（j） $ p（x）\ in \ mathbb {q} [x] $最多$ d $。特别是，当$ d = 1 $时，上面的序列是算术进程。在本文中，我们显示了上述此类数字$ n $的渐近密度。当$ d = 1 $时，渐近密度等于$ 1/（k-1）$。尽管共同差异$ r $是在上述结果中任意固定的，但我们还检查了$ r $未固定的情况。本文中的大多数结果是通过使用属于Hardy字段的功能来推广的。
By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $α\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^α}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^α}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, 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.
