George Bender | Searcharxiv

arXiv2025

On finitely many base $q$ expansions

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions.

math.DS↗

arXiv2023

On the cardinality and dimension of the slices of Okamoto's functions

The graphs of Okamoto's functions, denoted by $K_q$, are self-affine fractal curves contained in $[0,1]^2$, parameterised by $q \in (1,2)$. In this paper we consider the cardinality and dimension of the intersection of these curves with horizontal lines. Our first theorem proves that if $q$ is sufficiently close to $2$, then $K_q$ admits a horizontal slice with exactly three elements. Our second theorem proves that if a horizontal slice of $K_q$ contains an uncountable number of elements then it has positive Hausdorff dimension provided $q$ is in a certain subset of $(1,2)$. Finally, we prove that if $q$ is a $k$-Bonacci number for some $k \in \mathbb{N}_{\geq 3}$, then the set of $y \in [0,1]$ such that the horizontal slice at height $y$ has $(2m+1)$ elements has positive Hausdorff dimension for any $m \in \mathbb{N}$. We also show that, under the same assumption on $q$, there is some horizontal slice whose cardinality is countably infinite.

math.DS↗