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Takagi Function Identities on Dyadic Rationals

Laura Monroe

TL;DR

The work investigates identities for the Takagi function on dyadic rationals by tying dilations of $\tau$ to the number of $D$-nodes in divide-and-conquer binary trees. It develops recurrences and closed forms for the count $\delta(n)$ of $D$-nodes, then translates these into explicit Takagi-function formulas for points $\frac{r}{2^k}$ and into expressions for the binary Hamming weight $s_1(n)$. Key outcomes include recursive and closed forms for $\tau(\frac{r}{2^k})$, a Hamming-weight relation $s_1(n)$ in terms of $\delta(n)$, and multiple corollaries for cumulative digit sums (Trollope). The results illuminate the interplay between binary-tree balance, number-theoretic digit properties, and fractal functions, providing new identities and alternative proofs of classical theorems with potential applications in combinatorics and analysis.

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.

Takagi Function Identities on Dyadic Rationals

TL;DR

The work investigates identities for the Takagi function on dyadic rationals by tying dilations of to the number of -nodes in divide-and-conquer binary trees. It develops recurrences and closed forms for the count of -nodes, then translates these into explicit Takagi-function formulas for points and into expressions for the binary Hamming weight . Key outcomes include recursive and closed forms for , a Hamming-weight relation in terms of , and multiple corollaries for cumulative digit sums (Trollope). The results illuminate the interplay between binary-tree balance, number-theoretic digit properties, and fractal functions, providing new identities and alternative proofs of classical theorems with potential applications in combinatorics and analysis.

Abstract

The number of unbalanced interior nodes of divide-and-conquer trees on 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.

Paper Structure

This paper contains 8 sections, 25 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. The Takagi function and divide-and-conquer trees
  4. --nodes in divide-and-conquer trees
  5. Counting --nodes in a divide-and-conquer tree
  6. Counting --nodes on a divide-and-conquer tree in terms of Hamming weight
  7. From --nodes to the Takagi function
  8. Acknowledgments

Key Result

Lemma 2

Let $n, k, r, x$ be as in Notation not_n. Then $x=\frac{r}{2^k}$ and $1+x = \frac{n}{2^k}.$

Figures (1)

  • Figure 1: Divide-and-conquer dilations of the Takagi function on the dyadic rationals. Subfigures (\ref{['delta16']}), (\ref{['delta64']}) and (\ref{['delta256']}) show examples of the dilations $y=\frac{\delta(2^k+x)}{2^k}=\tau\left(\frac{x}{2^k}\right)$ from Theorem \ref{['takagi2']}, where $\delta(n)$ is the number of $D$-nodes on a divide-and-conquer tree on $n$ leaves. Here, $k=4,6,\text{ and }8$, and $x$ is an integer with $0 \le x \le 2^k$. These may be visually compared to Subfigure (\ref{['takagifig']}), showing the continuous, self-similar, nowhere-differentiable Takagi (blancmange) curve on $[0,1]$. (The blancmange curve image in Subfigure (\ref{['takagifig']}) is taken from Wiki Commons.)

Theorems & Definitions (44)

  • Lemma 2
  • Definition 3: Takagi function
  • Definition 4: $S$-nodes and $D$-nodes
  • Lemma 5
  • Definition 6: Divide-and-conquer tree
  • Theorem 8
  • Theorem 9
  • Corollary 10
  • proof
  • Theorem 11
  • ...and 34 more