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The Brjuno and Wilton Functions

Claire Burrin, Seul Bee Lee, Stefano Marmi

TL;DR

The paper unifies the Brjuno function $B$ and the Wilton function $W$ through the semi-Brjuno function $B_0$, showing that $B$ and $W$ correspond to the even/odd parts of $B_0$ up to uniform defects. It provides explicit formulas for the even/odd components, proves that $\Delta^+$ is $1/2$-Hölder while $\Delta^-$ exhibits popcorn-like discontinuities at rationals, and establishes two main results: a tight uniform approximation of $B$ and $W$ by $2B_0^+$ and $2B_0^-$, and a detailed regularity result for $\Delta^-$ with jumps at rationals. The work also extends the framework to complex Brjuno and Wilton functions, delivering explicit dilogarithm-based closed forms and showing that $\tfrac{1}{2}\mathrm{Im}(\mathcal{B}+\mathcal{W})$ converges to $B_0$ under tangential limits. Altogether, the results connect one-dimensional dynamical small-divisor phenomena with analytic number-theoretic divisor-sum structures in a cohesive, analyzable way.

Abstract

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $Δ^+(x) = B^+(x) - 2B_0^+(x)$ and $Δ^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

The Brjuno and Wilton Functions

TL;DR

The paper unifies the Brjuno function and the Wilton function through the semi-Brjuno function , showing that and correspond to the even/odd parts of up to uniform defects. It provides explicit formulas for the even/odd components, proves that is -Hölder while exhibits popcorn-like discontinuities at rationals, and establishes two main results: a tight uniform approximation of and by and , and a detailed regularity result for with jumps at rationals. The work also extends the framework to complex Brjuno and Wilton functions, delivering explicit dilogarithm-based closed forms and showing that converges to under tangential limits. Altogether, the results connect one-dimensional dynamical small-divisor phenomena with analytic number-theoretic divisor-sum structures in a cohesive, analyzable way.

Abstract

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function . Namely, and can be expressed in terms of the even and odd parts of , respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions and , the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

Paper Structure

This paper contains 5 sections, 8 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Even/odd parts and functional equations
  3. Proof of Theorem \ref{['thm:1']}
  4. Proof of Theorem \ref{['thm:2']}
  5. Complex Brjuno and Wilton functions

Key Result

Theorem 1.1

Let $B_0^\pm$ denote the even and odd parts of the semi-Brjuno function, i.e., $B_0^\pm(x) \coloneqq \frac{1}{2} (B_0(x) \pm B_0(-x)).$ Then there exist constants $C^\pm$ such that uniformly for all irrational numbers $x$.

Figures (4)

  • Figure 1: Plots of $B$ and $W$ at 10000 randomly chosen points
  • Figure 2: Plot of $B_0$ at 10000 randomly chosen points
  • Figure 3: Plots of $\Delta^+$ and $\Delta^-$ at 10000 randomly chosen points
  • Figure 4: The graph of $\Phi$ on $(0,\frac{1}{2})$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2: Functional equation for $W^-$
  • proof
  • Remark 2.3
  • Corollary 2.4: Functional equation for $\Delta^-$
  • proof
  • Remark 2.5
  • ...and 6 more