On Fourier asymptotics and effective equidistribution
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916522568450048 |
|---|---|
| author | Datta, Shreyasi Jana, Subhajit |
| author_facet | Datta, Shreyasi Jana, Subhajit |
| contents | We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis.
In particular, for any Borel probability measure $μ$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehatμ(m)| = O\left(X^{1/2-θ}\right)$ with $θ>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, and as well as, a sub-class of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $θ$ can be chosen arbitrarily small): there are measures $μ$ with $\widehatμ(ξ) = O\left(|ξ|^{-1/2+ε}\right)$ for which equidistribution fails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Fourier asymptotics and effective equidistribution Datta, Shreyasi Jana, Subhajit Dynamical Systems Number Theory 11J83, 28A80 We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure $μ$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehatμ(m)| = O\left(X^{1/2-θ}\right)$ with $θ>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, and as well as, a sub-class of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $θ$ can be chosen arbitrarily small): there are measures $μ$ with $\widehatμ(ξ) = O\left(|ξ|^{-1/2+ε}\right)$ for which equidistribution fails. |
| title | On Fourier asymptotics and effective equidistribution |
| topic | Dynamical Systems Number Theory 11J83, 28A80 |
| url | https://arxiv.org/abs/2407.11961 |