Shrinking targets versus recurrence: the quantitative theory

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