  We establish new recurrence and multiple recurrence results for a rather
large family $\mathcal{F}$ of non-polynomial functions which includes tempered
functions defined in [11], as well as functions from a Hardy field with the
property that for some $\ell\in \mathbb{N}\cup\{0\}$, $\lim_{x\to\infty
}f^{(\ell)}(x)=\pm\infty$ and $\lim_{x\to\infty }f^{(\ell+1)}(x)=0$. Among
other things, we show that for any $f\in\mathcal{F}$, any invertible
probability measure preserving system $(X,\mathcal{B},\mu,T)$, any
$A\in\mathcal{B}$ with $\mu(A)>0$, and any $\epsilon>0$, the sets of returns $$
R_{\epsilon, A}= \big\{n\in\mathbb{N}:\mu(A\cap T^{-\lfloor
f(n)\rfloor}A)>\mu^2(A)-\epsilon\big\} $$ and $$
  R^{(k)}_{A}= \big\{ n\in\mathbb{N}: \mu\big(A\cap T^{\lfloor
f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor
f(n+k)\rfloor}A\big)>0\big\} $$ possess somewhat unexpected properties of
largeness; in particular, they are thick, i.e., contain arbitrarily long
intervals.
