Shrinking targets versus recurrence: the quantitative theory

Abstract

Let $X=[0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\ddot{I}ˆ:\mathbb{N}\to {\mathbb{R}}_{\ge 0}$, $x\in X$, and $N\in \mathbb{N}$ we consider the recurrence counting function \[ R(x,N;T,ψ) := \#\{1\leq n\leq N: d(T^n x, x) < ψ(n)\}. \] We show that for any $\epsilon >0$ we have \[ R(x,N;T,ψ) = Ψ(N)+O\left(Ψ^{1/2}(N) \ (\logΨ(N))^{3/2+\varepsilon}\right) \] for $\hat{I}¼$-almost all $x\in X$ and for all $N\in \mathbb{N}$, where $\hat{I}¨(N):=2{\sum }_{n=1}^N\ddot{I}ˆ(n)$. We also prove a generalization of this result to higher dimensions.