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

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 $k\in {\mathbb{N}}_{\ge 3}$, 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.