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arXiv · 2409.20401

Regularity properties of the $\alpha$-Wilton functions

Ayreena Bakhtawar·Carlo Carminati·Seul Bee Lee
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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}).

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Ayreena Bakhtawar, Carlo Carminati, Seul Bee Lee. 2024-09-30. Regularity properties of the $\alpha$-Wilton functions. https://arxiv.org/abs/2409.20401

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