论文标题
Hofstadter的$ H $序列中的一个隐藏信号
A hidden signal in Hofstadter's $H$ sequence
论文作者
论文摘要
hofstadter $ h $序列由$ h（1）= 1 $和$ h（n）= n-h（h（h（n-1）））$定义，$ n> 1 $。如果$α$是$ x^3+x = 1 $的真实根，我们表明数字$αH（n）\ mod 1 $不是均匀分布在$ [0,1] $上，而是收敛到我们认为连续但不是可区分的分布。这是由于发现Steinerberger的发现，他发现了ULAM序列具有相似行为的真实数字。我们的结果与以下事实有关：从线性复发定义的某个序列$ h_n = h_ {n-1}+h_ {n-3} $具有属性$ \ | x h_n \ | \ rightArrow 0 $正好适合$ x \ in \ mathbb {z} [α] $，这是我们查询整数的一般线性复发序列的现象。
The Hofstadter $H$ sequence is defined by $H(1) = 1$ and $H(n) = n-H(H(H(n-1)))$ for $n > 1$. If $α$ is the real root of $x^3+x=1$ we show that the numbers $αH(n) \mod 1$ are not uniformly distributed on $[0,1]$, but converge to a distribution we believe is continuous but not differentiable. This is motivated by a discovery of Steinerberger, who found a real number with similar behavior for the Ulam sequence. Our result is related with the fact that a certain sequence defined from the linear recurrence $h_n=h_{n-1}+h_{n-3}$ has the property $\|x h_n\| \rightarrow 0$ precisely for $x \in \mathbb{Z}[α]$, a phenomenon we inquire for general linear recurrent sequences of integers.
