Small gaps in the Ulam sequence

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 $n \in \mathbb{N}$ $$ \min_{1 \leq k \leq n} \frac{a_{k+1}}{a_k} \leq 1 + c\frac{\log{n}}{n}.$$