Small gaps in the Ulam sequence
François Clément, Stefan Steinerberger
TL;DR
The paper analyzes the Ulam sequence, defined by $a_1=1$, $a_2=2$, and $a_{n+1}$ as the smallest integer with a unique representation as a sum of two distinct earlier terms. It introduces a matrix-based framework using three recurrences (Eggleton, Type I, Type II) encoded by matrices $T_1,T_2,T_3$, and employs submultiplicativity and joint spectral radius ideas to study growth and sums. It proves an improved upper bound $a_n \leq 1.454^n$ for large $n$ and establishes a small-gap bound $\min_{1 \leq k \leq n} \frac{a_{k+1}}{a_k} \leq 1 + c \frac{\log n}{n}$ with $c=7$, while acknowledging limitations of the method for proving subexponential growth below certain thresholds. The results advance the rigorous understanding of Ulam-type sequences by connecting combinatorial structure with spectral-analytic techniques and highlighting potential obstructions like clumped growth patterns.
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 $n \in \mathbb{N}$ $$ \min_{1 \leq k \leq n} \frac{a_{k+1}}{a_k} \leq 1 + c\frac{\log{n}}{n}.$$