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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We present a survey of ergodic theorems for actions of algebraic andarithmetic groups recently established by the authors, as well as some of theirapplications. Our approach is based on spectral methods employing the unitaryrepresentation theory of the groups involved. This allows the derivation ofergodic theorems with a rate of convergence, an important phenomenon which doesnot arise in classical ergodic theory. Quantitative ergodic theorems give riseto new and previously inaccessible applications, and we demonstrate theremarkable diversity of such applications by presenting severalnumber-theoretic results. These include, in particular, general uniform errorestimates in lattice points counting problems, explicit estimates in siftingproblems for almost-prime points on symmetric varieties, bounds for exponentsof intrinsic Diophantine approximation, and results on fast distribution ofdense orbits on homogeneous spaces.

Quantitative ergodic theorems and their number-theoretic applications