Number theoretic applications of a class of Cantor series fractal functions. I

Abstract

Suppose that ${{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}$ and x = E ₀.E ₁ E ₂ · · · is the P-Cantor series expansion of ${x \in \mathbb{R}}$. We define

$$\psi_{P,Q}(x) := {\sum_{n=1}^{\infty}} \frac{{\rm min}(E_n, q_{n}-1)}{q_1 \cdots q_n}.$$

The functions ${\psi_{P,Q}}$ are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of ${\psi_{P,Q}}$ restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [29] is improved.

Properties of the functions ${\psi_{P,Q}}$ are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of ${\psi_{P,Q}}$ on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures ${\mu_1}$ and ${\mu_2}$ on ${{\mathbb{N}_2^\mathbb{N}}}$, we study which of these properties ${\psi_{P,Q}}$ satisfies for ${\mu_1 \times \mu_2}$-almost every (P,Q) ${{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}$. Related classes of random fractals are also studied.