Evaluating the Fabius function

TL;DR

This paper explores the Fabius function, a continuous extension of the Thue-Morse sequence, providing a method to precisely compute its values at dyadic rationals.

Contribution

It introduces a technique to exactly evaluate the Fabius function at points that are ratios of positive integers and powers of two.

Findings

Exact values of the Fabius function at dyadic rationals are determined.

The method enhances understanding of the function's behavior at specific rational points.

The work bridges discrete sequences and continuous functions through explicit computation.

Abstract

The Thue-Morse sequence (1, -1, -1, 1, -1, 1, 1, ...) can in a sense be naturally extended to a continuous function f called the Fabius function. It is shown how to determine the exact value of f(x) whenever x is the ratio between a positive integer and a power of 2.