A Hidden Signal in the Ulam sequence

Abstract

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, \dots$$ Ulam remarked that understanding the sequence, which has been described as 'quite erratic', seems difficult and indeed nothing is known. We report the empirical discovery of a surprising global rigidity phenomenon: there seems to exist a real $\alpha \sim 2.5714474995\dots$ such that $$\left\{\alpha a_n: n\in \mathbb{N}\right\} \quad \mbox{mod}~2\pi \quad \mbox{generates an absolutely continuous \textit{non-uniform} measure}$$ supported on a subset of $\mathbb{T}$. Indeed, for the first $10^7$ elements of Ulam's sequence, $$ \cos{\left( 2.5714474995~ a_n\right)} < 0 \qquad \mbox{for all}~a_n \notin \left\{2, 3, 47, 69\right\}.$$ The same phenomenon arises for some other initial conditions $a_1, a_2$: the distribution functions look very different from each other and have curious shapes. A similar but more subtle phenomenon seems to arise in Lagarias' variant of MacMahon's 'primes of measurement' sequence.