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

<p dir="ltr">The graphs of Okamoto’s functions, denoted by <i>K</i>_<em>q</em>​, are self-affine fractal curves contained in [0,1]², parameterised by <i>q</i>∈(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 <i>q</i> is sufficiently close to 2, then <i>K</i>_<em>q</em>_​ admits a horizontal slice with exactly three elements. Our second theorem proves that if a horizontal slice of <i>K</i>_<em>q</em>​ contains an uncountable number of elements then it has positive Hausdorff dimension provided <i>q</i> is in a certain subset of (1,2). Finally, we prove that if <i>q</i> is a <i>k</i>-Bonacci number for some <i>k </i>∈ N≥3​, then the set of <i>y </i>∈ [0,1] such that the horizontal slice at height <i>y</i> has (2<i>m </i>+ 1) elements has positive Hausdorff dimension for any <i>m </i>∈ N. We also show that, under the same assumption on <i>q</i>, there is some horizontal slice whose cardinality is countably infinite.</p>