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 $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function [ R(x,N;T,\psi) := #{1\leq n\leq N: d(T^n x, x) < \psi(n)}. ] We show that for any $\varepsilon > 0$ we have [ R(x,N;T,\psi) = \Psi(N)+O\left(\Psi^{1/2}(N) \ (\log\Psi(N))^{3/2+\varepsilon}\right) ] for $\mu$-almost all $x\in X$ and for all $N\in\mathbb N$, where $\Psi(N):= 2 \sum_{n=1}^N \psi(n)$. We also prove a generalization of this result to higher dimensions.