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∈(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 ∈ N≥3​, then the set of y ∈ [0,1] such that the horizontal slice at height y has (2m + 1) elements has positive Hausdorff dimension for any m ∈ N. We also show that, under the same assumption on q, there is some horizontal slice whose cardinality is countably infinite.