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

## TL;DR

This paper investigates the conditions under which points in certain dynamical systems will eventually always hit shrinking targets, establishing zero-one laws for systems with near independent target translates like Bernoulli schemes and the Gauss map.

## Contribution

It provides new zero-one laws for eventually always hitting points in rapidly mixing systems with shrinking targets, based on the targets' shrinking rates and near independence.

## Key findings

- Zero-one laws depend on the shrinking rate of targets.
- Systems with near independence exhibit sharp thresholds for hitting behavior.
- Results apply to Bernoulli schemes and the Gauss map.

## 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).