On fractal patterns in Ulam words

Abstract

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.