Small gaps in the Ulam sequence

TL;DR

This paper improves the upper bound on the growth rate of the Ulam sequence and proves that small gaps must exist between its terms, revealing underlying structure in its irregular pattern.

Contribution

It provides the first improved upper bound on the sequence's growth and establishes the existence of small gaps, advancing understanding of its structure.

Findings

Improved upper bound on the growth rate of the Ulam sequence.

Proof that small gaps between terms must exist.

Quantitative bound involving logarithmic growth.

Abstract

The Ulam sequence, described by Stanislaw Ulam in the 1960s, starts $1,2$ and then iteratively adds the smallest integer that can be uniquely written as the sum of two distinct earlier terms: this gives $1,2,3,4,6,8,11,\dots$. Already in 1972 the great French poet Raymond Queneau wrote that it `gives an impression of great irregularity'. This irregularity appears to have a lot of structure which has inspired a great deal of work; nonetheless, very little is rigorously proven. We improve the best upper bound on its growth and show that at least some small gaps have to exist: for some $c>0$ and all $n \in \mathbb{N}$ $$ \min_{1 \leq k \leq n} \frac{a_{k+1}}{a_k} \leq 1 + c\frac{\log{n}}{n}.$$