On fractal patterns in Ulam words

TL;DR

This paper explores the structure of Ulam words, especially those with two '1's, revealing their intricate properties, algorithms for recognition, and their connection to fractal patterns like the Sierpinski gasket.

Contribution

It provides a complete characterization of two-'1' Ulam words, introduces a logarithmic-time recognition algorithm, and constructs fractals based on their hierarchical structure.

Findings

Complete description of two-'1' Ulam words

Logarithmic-time algorithm for recognition

Construction of self-similar fractals from 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.