[Paper Review] On the Hausdorff dimension of some sets of numbers defined through the digits of their $Q$-Cantor series expansions

Dylan Airey, Bill Mance|arXiv (Cornell University)|Jul 3, 2014

Mathematical Dynamics and Fractals11 references4 citations

TL;DR

This paper computes the Hausdorff dimension of difference sets defined by various normality properties in $Q$-Cantor series expansions, showing that all nontrivial such sets have full Hausdorff dimension (1) for basic sequences $Q$ that are infinite in limit and fully divergent, except for $\mathscr{N}(Q)\setminus\mathscr{DN}(Q)$, which also has full dimension under stronger conditions. The results extend classical normal number theory to generalized Cantor series expansions using dimension-theoretic methods.

ABSTRACT

Following in the footsteps of P. Erdős and A. Rényi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we consider difference sets associated with various notions of normality and sets of numbers with a prescribed range of digits.

Motivation & Objective

• To determine the Hausdorff dimension of sets of real numbers whose $Q$-Cantor series digits satisfy specific statistical properties.
• To extend classical results on normal numbers in base-$b$ expansions to the more general setting of $Q$-Cantor series expansions.
• To resolve open questions about the dimension of difference sets between $Q$-normal, $Q$-ratio normal, and $Q$-distribution normal sets.
• To establish conditions under which the set $\mathscr{N}(Q)\setminus\mathscr{DN}(Q)$ has full Hausdorff dimension, despite not being full measure.

Proposed method

• Uses the framework of $Q$-Cantor series expansions, where $Q = (q_n)$ is a basic sequence with $q_n \geq 2$, to generalize base-$b$ expansions.
• Defines $Q$-normality of order $k$ via asymptotic equidistribution of digit blocks of length $k$ in the expansion.
• Applies techniques from metric number theory and fractal geometry, particularly the construction of homogeneous Moran sets to bound Hausdorff dimension from below.
• Employs a limiting argument involving $Q_n^{(k)} = \sum_{j=1}^n \frac{1}{q_j q_{j+1} \cdots q_{j+k-1}}$ to analyze frequency of digit blocks.
• Constructs explicit subsets of $\mathscr{W}_Q(S)$, the set of numbers with digit set $S$ in their $Q$-expansion, using sparse index sets to control dimension.
• Uses the formula $\dim_H(C) \geq \liminf_{k\to\infty} \frac{\log(n_1 \cdots n_k)}{-\log(c_1 \cdots c_{k+1} n_{k+1})}$ to estimate dimension of constructed Moran sets.

Experimental results

Research questions

• RQ1 What is the Hausdorff dimension of the set $\mathscr{N}(Q) \setminus \mathscr{DN}(Q)$ for basic sequences $Q$ that are infinite in limit and fully divergent?
• RQ2 Do all nontrivial difference sets formed from $\mathscr{N}(Q)$, $\mathscr{RN}(Q)$, and $\mathscr{DN}(Q)$ have full Hausdorff dimension when $Q$ is infinite in limit and fully divergent?
• RQ3 Can the full dimension result for $\mathscr{N}(Q) \setminus \mathscr{DN}(Q)$ be extended to all such $Q$, or only to a restricted subclass?
• RQ4 What conditions on $Q$ ensure that $\bigcap_{j=\ell}^\infty \mathscr{N}_j(Q) \setminus \bigcup_{j=1}^{\ell-1} \mathscr{N}_j(Q)$ has full Hausdorff dimension?
• RQ5 Is there a necessary and sufficient condition for full Hausdorff dimension of sets like $\mathscr{DN}(Q) \setminus \mathscr{N}(Q)$, analogous to Schmidt's conditions in classical theory?

Key findings

• For every basic sequence $Q$ that is infinite in limit and fully divergent, all nontrivial difference sets formed from $\mathscr{N}(Q)$, $\mathscr{RN}(Q)$, and $\mathscr{DN}(Q)$ have Hausdorff dimension 1.
• The set $\mathscr{N}(Q) \setminus \mathscr{DN}(Q)$ has full Hausdorff dimension (i.e., dimension 1) when $Q$ satisfies additional growth conditions, including $k$-divergence for $k \geq \ell$ and specific asymptotic behavior of $q_n$.
• The set $\mathscr{W}_Q(S)$, of numbers whose digits lie in a fixed set $S \subset \mathbb{N}_0$, has Hausdorff dimension equal to $\gamma = \lim_{k\to\infty} \frac{\log \#(S \cap \{0, \dots, q_k - 2\})}{\log q_k}$, provided $Q$ is infinite in limit.
• A construction of a homogeneous Moran set with dimension matching $\gamma$ proves the lower bound for $\dim_H(\mathscr{W}_Q(S))$, and the upper bound follows from containment in a controlled set.
• The paper shows that $\dim_H(\mathscr{DN}(Q) \setminus \mathscr{N}(Q)) = 1$ and $\dim_H(\mathscr{DN}(Q) \setminus \mathscr{RN}(Q)) = 1$ for infinite-in-limit, fully divergent $Q$, confirming full dimension for these difference sets.
• The results generalize earlier findings by showing that full dimension is robust across various combinations of normality notions, even when the sets are not of full Lebesgue measure.