Takagi Function Identities on Dyadic Rationals

TL;DR

This paper explores identities involving the Takagi function at dyadic rationals and relates them to the Hamming weight of integers, revealing new connections in divide-and-conquer tree structures.

Contribution

It introduces novel identities linking the Takagi function with Hamming weights and unbalanced nodes in divide-and-conquer trees on dyadic rationals.

Findings

Identifies a sequence of dilations of the Takagi function related to tree structures.

Derives new identities connecting the Takagi function with Hamming weights.

Establishes mathematical relationships between these concepts.

Abstract

The number of unbalanced interior nodes of divide-and-conquer trees on $n$ leaves is known to form a sequence of dilations of the Takagi function on dyadic rationals. We use this fact to derive identities on the Takagi function and on the Hamming weight of an integer in terms of the Takagi function.