On the dimension theory of Okamoto's function

TL;DR

This paper analyzes the fractal dimensions of Okamoto's function, computing various dimensions of its graph and level sets for typical parameters, revealing their typical size and structure.

Contribution

It provides the first detailed dimension analysis of Okamoto's function, including Hausdorff, box-counting, and Assouad dimensions, and characterizes the typical size of its level sets.

Findings

Hausdorff, box-counting, and Assouad dimensions of the graph are computed for typical parameters.

An upper bound on the dimension of level sets is established.

For typical parameters, the dimension of almost every level set attains the upper bound.

Abstract

In this paper, we investigate the dimension theory of the one parameter family of Okamoto's function. We compute the Hausdorff, box-counting and Assouad dimensions of the graph for a typical choice of parameter. Furthermore, we study the dimension of the level sets. We give an upper bound on the dimension of every level set, and we show that for a typical choice of parameters this value is attained for Lebesgue almost every level sets.