Cibles r\'etr\'ecissantes de rayon $n^{-\frac{1}{d}}$ : propri\'et\'e du logarithme

TL;DR

This paper investigates the logarithm property for translations on the d-dimensional torus, establishing criteria based on Diophantine conditions and providing counterexamples and examples in two dimensions.

Contribution

It introduces new Diophantine-based criteria for the logarithm property and constructs counterexamples and positive cases in two-dimensional tori.

Findings

Irrational translations on $ ext{T}^1$ always have the logarithm property.

Counterexamples exist in $ ext{T}^2$ with vectors of small Diophantine type.

Logarithm property can hold for Liouvillian vectors in two dimensions.

Abstract

A translation on the d-dimensional torus $\mathbb{T}^d$ has the logarithm property if the Shrinking Target Property holds for the sequence of balls with radius $n^{-\frac{1}{d}}$. On $\mathbb{T}^1$ every irrational translations has this property. In higher dimension, we give criterions based upon an unusual Diophantine type. In the two dimensional torus, we construct counter examples to the logarithm with translations vectors of arbitrary small Diophantine type and example of logarithm property with Liouvillian vector