Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularity properties of $k$-Brjuno and Wilton functions

Seul Bee Lee, Stefano Marmi, Izabela Petrykiewicz, Tanja I. Schindler

Abstract

We study functions related to the classical Brjuno function, namely $k$-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $α$-continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations.

Regularity properties of $k$-Brjuno and Wilton functions

Abstract

We study functions related to the classical Brjuno function, namely -Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the -continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations.

Paper Structure

This paper contains 28 sections, 29 theorems, 293 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Brjuno function
  3. $k$-Brjuno functions
  4. $\alpha$-continued fractions
  5. Wilton function
  6. BMO properties and complex $k$-Brjuno and Wilton functions.
  7. Structure of the paper
  8. $\textnormal{BMO}$ properties of the real $k$-Brjuno and the real Wilton function
  9. Proof that the real $k$-Brjuno functions are $\textnormal{BMO}$ functions
  10. Proofs of the $\textnormal{BMO}$ properties of the Wilton function
  11. Behaviour of the truncated real Brjuno function and the truncated real Wilton function
  12. Complex continued fractions
  13. Complexification of the operators $T_k$ and $S$
  14. Hyperfunctions and extensions to spaces of complex analytic functions of the operators $T_k$ and $S$
  15. Hyperfunctions
  16. ...and 13 more sections

Key Result

Theorem 1.1

Figures (9)

  • Figure 1: Numerical computation of the Brjuno function $B$ (left) and of the Wilton function $W$ (right) when $\alpha=1$. The asymmetric logarithmic singularities at rational points provide an intuitive justification for $W$ not belonging to the BMO space, see Section \ref{['sec:BMO']}.
  • Figure 2: Numerical computations of $W_\alpha$ for $\alpha=1/2$ (upper left), $\alpha=\frac{\sqrt{5}-1}{2}$ (upper right), $\alpha=e-2$ (lower left) and $\alpha=0.9$ (lower right).
  • Figure 3: The graph of $A_{1/2}$ and $A_\alpha$ for $\alpha\in[\frac{1}{2},g]$, where $g=\frac{\sqrt{5}-1}{2}$.
  • Figure 4: Graph of $A_\alpha$ when $\alpha>g$.
  • Figure 5: The sets $D(m_1,\cdots,m_r)$. The left figure is the partition of $D$ by $D(n)$. The right figure is of the sets $D(1,n)$ in $D(1)$.
  • ...and 4 more figures

Theorems & Definitions (56)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4: MMY1
  • Proposition 2.5
  • Remark 2.6
  • proof : Proof of Proposition \ref{['prop: kBrjuno equiv to prop2.3']}
  • Remark 2.7
  • Proposition 2.8
  • ...and 46 more