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

TL;DR

This paper investigates the size and fractal dimension of horizontal slices of Okamoto's fractal curves, revealing conditions under which slices have specific cardinalities and positive Hausdorff dimension, especially near $q=2$ and for $k$-Bonacci numbers.

Contribution

It provides new results on the cardinality and Hausdorff dimension of slices of Okamoto's functions, including cases with exactly three points, uncountably many points, and positive dimension for certain parameters.

Findings

Existence of horizontal slices with exactly three points near $q=2$.

Uncountable slices have positive Hausdorff dimension under certain conditions.

Slices with $(2m+1)$ points have positive Hausdorff dimension for $k$-Bonacci $q$.

Abstract

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.