English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the distribution of the sequence of continued fraction convergents$p_n/q_n$ to a random irrational number in the group$\mathrm{SL}(2,\mathbb{Z}_m)$, and in particular the distribution of $p_n\pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$. Improving the strong lawof large numbers due to Sz\"usz, Moeckel, Jager and Liardet, we establish thecentral limit theorem and the law of the iterated logarithm, as well as theweak and the almost sure invariance principles. As an application, we find thelimit distribution of the maximum and the minimum of the Birkhoff sum for theirrational rotation with the indicator of an interval as test function. We alsocompute the normalizing constant in a classical limit law for the same Birkhoffsum due to Kesten, and dispel a misconception about its dependence on the testinterval.

Equidistribution of continued fraction convergents in $\mathrm{SL}(2,\mathbb{Z}_m)$ with an application to local discrepancy