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.