An effective estimate for the Lebesgue measure of preimages of iterates of the Farey map

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• 作者：Byron Heersink ^{heersin2@illinois.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Farey map ; Continued fractions ; Stern&ndash ; Brocot sequence ; Karamata's Tauberian theorem
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：19 March 2016
• 年：2016
• 卷：291
• 期：Complete
• 页码：621-634
• 全文大小：329 K

文摘

Using techniques from infinite ergodic theory, Kesseböhmer and Stratmann determined the asymptotic behavior of the Lebesgue measure of sets of the form F⁻ⁿ[α,β]$F^{-n}[\alpha ,\beta ]$, where [α,β]⊆(0,1]$[\alpha ,\beta ]\subseteq (0,1]$ and F is the Farey map. In this paper, we provide an effective version of this result, employing mostly basic properties of the transfer operator of the Farey map and an application of Freud's effective version of Karamata's Tauberian theorem.