Regularity properties of the $\hat{I}\pm$-Wilton functions

Abstract

The aim of this article is to study the regularity properties of the Wilton functions $W_{\hat{I}}\pm$ associated with $\hat{I}\pm$-continued fractions. We prove that the Wilton function is BMO for $\hat{I}\pm \in [1-g,g]$ (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 $\hat{I}\pm$ for which $W_{\hat{I}}\pm$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $\hat{I}\pm$-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, $\hat{I}\pm =1/2$ (\cite{MaMoYo_97}) and $\hat{I}\pm \in [\frac{1}{2},g]$ (\cite{LeMar_24}).

21 pages, 5 figures