The joint distribution of binary and ternary digits sums

TL;DR

This paper proves that the sum-of-digits functions in bases 2 and 3 take on almost all pairs of values, showing that generalized collisions occur infinitely often for scaled sums, extending a long-standing conjecture.

Contribution

It generalizes the known collision result to show that the pairs of sum-of-digits values in bases 2 and 3 are asymptotically dense in N^2, and establishes infinitely many solutions for scaled equations.

Findings

$(s_2(n),s_3(n))$ attains almost all pairs in $ ^2$

There are infinitely many solutions to $as_2(n)=bs_3(n)$ for positive integers $a,b$

Confirmed the conjecture about the density of sum-of-digits pairs in different bases

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 $n$.