论文标题
$ x_n = f（n）^n $的序列的密度[0,1]
Density of sequences of the form $x_n=f(n)^n$ in [0,1]
论文作者
论文摘要
2013年，Strauch询问了如何从三角函数（例如$ x_n =（\ cos n）^n $分发的三角函数定义的各种实数序列。 Strauch的询问是由几个这样的分配结果激发的。例如，卢卡（Luca）证明了$ x_n =（\cosαn）^n \ pmod 1 $在$ [0,1] $中的序列$ [0,1] $，对于任何固定的实际数量$α$，因此$α/π$是不合理的。在这里，我们将卢卡的结果推广到表单$ x_n = f（n）^n \ pmod 1 $的其他序列。我们还检查了集合$ | \ {n \ leq n：r <| \ cos（nπα）|^n \} | $的大小，其中$ 0 <r <1 $和$α$是固定的，使得$α/π$是不合理的。
In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as $x_n=(\cos n)^n$ distributed themselves$\pmod 1$. Strauch's inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence $x_n=(\cos αn)^n\pmod 1$ is dense in $[0,1]$ for any fixed real number $α$ such that $α/π$ is irrational. Here we generalise Luca's results to other sequences of the form $x_n=f(n)^n\pmod 1$. We also examine the size of the set $|\{n\leq N:r<|\cos(nπα)|^n\}|$ where $0<r<1$ and $α$ are fixed such that $α/π$ is irrational.
