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 $μ^{(r)}(d)$ 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 $μ^{(r)}$ 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 $μ^{(r)}$ as the number of blocks goes to infinity. We show that, up to renormalization, $μ^{(r)}$ 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 $φ$-mixing process defined on the $b$-adic integers.

Figures (10)

• Figure 1: Examples of the decomposition in blocks in decimal and binary bases. On the left-hand side, $\rho(r)=7$. On the right-hand side, $\rho(r)=9$.
• Figure 2: Behavior of $T$ on the Rokhlin tower of order $0$.
• Figure 3: How to construct the tower of order $1$ of $T$ from the tower of order $0$ in base $b=4$.
• Figure 4: Visual description of $V_{\ell-1}$, $V_{\ell}$ and $V_{\ell}\setminus V_{\ell-1}$.
• Figure 5: Values of $\Delta^{(1)}$ on the levels of the Rokhlin tower of order $0$, and the corresponding $\mathbb{P}$-measures.

Theorems & Definitions (48)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Corollary 2.3
• Remark 1
• Lemma 2.4