Mathematical Research Letters

Volume 30 (2023)

Number 3

Zero–one laws for eventually always hitting points in rapidly mixing systems

Pages: 765 – 805

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n3.a7

Authors

Dmitry Kleinbock (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Ioannis Konstantoulas (Departement of Mathematics, Uppsala University, Uppsala, Sweden)

Florian K. Richter (Institute of Mathematics, École Polytechnique Fédéral de Lausanne, Switzerland)

Abstract

In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co‑null set respectively).

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This material is based upon work supported by the National Science Foundation under Grant Number DMS-1641020 and DMS-1926686. The first-named author was supported in part by NSF grants DMS-1600814 and DMS-1900560.

Received 3 December 2020

Received revised 23 January 2023

Accepted 7 February 2023

Published 15 December 2023