The joint distribution of binary and ternary digits sums
Michael Drmota, Lukas Spiegelhofer
TL;DR
This work resolves the joint behavior of two sum-of-digits functions, $s_2(n)$ and $s_3(n)$, by proving that the pair $(s_2(n),s_3(n))$ attains almost all values in $ abla^2\mathbb{N}$ asymptotically, and as a corollary establishes infinitely many generalized collisions: for any positive integers $(a,b)$, $a s_2(n)=b s_3(n)$ has infinitely many solutions. The authors develop two core propositions giving sharp exponential-sum bounds and a Diophantine toolkit (including Baker-type bounds and p-adic subspace theorems) to control digit-structure interactions across bases $2$ and $3$. A digit-elimination approach combined with van der Corput-type estimates yields strong bounds for small and large $K$, while a key lemma LeKey shows near-independence of digits, enabling a CLT-type joint Gaussian limit for $(s_2(n),s_3(n))$ and establishing a local limit theorem. The results connect base-$2$ and base-$3$ digit statistics to broader questions about factorial base representations, Catalan-number divisibilities, and ergodic-type conjectures, thereby enriching the understanding of multi-base digit expansions and their collisions.
Abstract
We consider the sum-of-digits functions $s_2$ and $s_3$ in bases $2$ and $3$. These functions just return the minimal numbers of powers of two (resp. three) needed in order to represent a nonnegative integer as their sum. A result of the second author states that there are infinitely many \emph{collisions} of $s_2$ and $s_3$, that is, positive integers $n$ such that \[s_2(n)=s_3(n).\] This resolved a long-standing folklore conjecture. In the present paper, we prove a strong generalization of this statement, stating that $(s_2(n),s_3(n))$ attains almost all values in $\mathbb N^2$, in the sense of asymptotic density. In particular, this yields \emph{generalized collisions}: for any pair $(a,b)$ of positive integers, the equation \[as_2(n)=bs_3(n)\] admits infinitely many solutions in $n$.