Regularity properties of the $α$-Wilton functions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910625386463232 |
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| author | Bakhtawar, Ayreena Carminati, Carlo Lee, Seul Bee |
| author_facet | Bakhtawar, Ayreena Carminati, Carlo Lee, Seul Bee |
| contents | The aim of this article is to study the regularity properties of the Wilton functions $W_α$ associated with $α$-continued fractions. We prove that the Wilton function is BMO for $α\in[1-g,g]$ (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 $α$ for which $W_α$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $α$-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, $α=1/2$ (\cite{MaMoYo_97}) and $α\in[\frac{1}{2},g]$ (\cite{LeMar_24}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity properties of the $α$-Wilton functions Bakhtawar, Ayreena Carminati, Carlo Lee, Seul Bee Dynamical Systems Number Theory 0B70, 11J70, 11K50 The aim of this article is to study the regularity properties of the Wilton functions $W_α$ associated with $α$-continued fractions. We prove that the Wilton function is BMO for $α\in[1-g,g]$ (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 $α$ for which $W_α$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $α$-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, $α=1/2$ (\cite{MaMoYo_97}) and $α\in[\frac{1}{2},g]$ (\cite{LeMar_24}). |
| title | Regularity properties of the $α$-Wilton functions |
| topic | Dynamical Systems Number Theory 0B70, 11J70, 11K50 |
| url | https://arxiv.org/abs/2409.20401 |