A central limit theorem for the variation of the sum of digits

Abstract

We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $\mu^{(r)}(d)$ as the density of integers $n\in \mathbb{N}$ for which the sum of digits increases by $d$ when we add $r$ to $n$. We give a probabilistic interpretation of $\mu^{(r)}$ on the probability space given by the group of $b$-adic integers equipped with the normalized Haar measure. We split the base-$b$ expansion of the integer $r$ into so-called "blocks", and we consider the asymptotic behaviour of $\mu^{(r)}$ as the number of blocks goes to infinity. We show that, up to renormalization, $\mu^{(r)}$ converges to the standard normal law as the number of blocks of $r$ grows to infinity. We provide an estimate of the speed of convergence. The proof relies, in particular, on a $\phi$-mixing process defined on the $b$-adic integers.