Regularity properties of the $\alpha$-Wilton functions

TL;DR

This paper investigates the regularity of Wilton functions linked to $eta$-continued fractions, establishing their bounded mean oscillation (BMO) properties within specific parameter ranges and demonstrating the optimality of these results.

Contribution

It extends previous work by precisely characterizing the BMO regularity of Wilton functions for all relevant $eta$ parameters, including boundary cases.

Findings

Wilton functions are BMO for $eta otin$ boundary neighborhoods of $g$

The BMO property is optimal and fails near boundary points

The proof uses the concept of 'matching' in $eta$-continued fractions

Abstract

The aim of this article is to study the regularity properties of the Wilton functions $W_\alpha$ associated with $\alpha$-continued fractions. We prove that the Wilton function is BMO for $\alpha\in[1-g,g]$ (where $g:=\frac{\sqrt{5}-1}{2}$ denotes the golden number), and we show that this result is optimal, since we find that on any left neighbourhood of $1-g$ and on any right neighbourhood of $g$ there are values $\alpha$ for which $W_\alpha$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $\alpha$-continued fractions called ``matching''. Our results complete those of Marmi--Moussa--Yoccoz (1997) and of Lee--Marmi--Petrykiewicz--Schindler (2024), where it is proven that Wilton function is BMO for, respectively, $\alpha=1/2$ (\cite{MaMoYo_97}) and $\alpha \in[\frac{1}{2},g]$ (\cite{LeMar_24}).