论文标题
关于乌拉姆单词的分形图案
On fractal patterns in Ulam words
论文作者
论文摘要
ulam单词是二进制单词递归定义的，如下所示：长度 - $ 1 $ ulam单词是$ 0 $和$ 1 $，而二进制$ n $的二进制单词是ulam时ulam当且仅当它以唯一形式表达为两个短而独特的ulam单词的串联。我们在包含两个$ 1 $的ulam词中发现，充分描述并证明了一个令人惊讶的丰富结构。特别是，这导致对此类单词的完整描述和对数时间算法，以确定带有两个$ 1 $的二进制单词是否为ulam。在此过程中，我们发现了微妙的奇偶校验和双层性能，以及在两个$ 1 $'s之外的$ 0 $ 0 $ 0 $ 0 $ 0的范围。我们还表明，两个$ 1 $之间的数字$ y $ 0 $ y $ y $ y $ 0 $的ulam单词具有复杂的基于张量的层次结构结构，该结构由$ y $的算术属性确定。这使我们能够构建一个无限的基于ULAM字的分形，该家族由$ 2 $ -ADIC的整数索引，其中包含外向Sierpinski垫圈作为特殊情况。
Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words. We discover, fully describe, and prove a surprisingly rich structure already in the set of Ulam words containing exactly two $1$'s. In particular, this leads to a complete description of such words and a logarithmic-time algorithm to determine whether a binary word with two $1$'s is Ulam. Along the way, we uncover delicate parity and biperiodicity properties, as well as sharp bounds on the number of $0$'s outside the two $1$'s. We also show that sets of Ulam words indexed by the number $y$ of $0$'s between the two $1$'s have intricate tensor-based hierarchical structures determined by the arithmetic properties of $y$. This allows us to construct an infinite family of self-similar Ulam-word-based fractals indexed by the set of $2$-adic integers, containing the outward Sierpinski gasket as a special case.
