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 $ellin mathbb{N}cup{0}$, $lim_{xtoinfty }f^{(ell)}(x)=pminfty$ and $lim_{xtoinfty }f^{(ell+1)}(x)=0$. Among other things, we show that for any $finmathcal{F}$, any invertible probability measure preserving system $(X,mathcal{B},mu,T)$, any $Ainmathcal{B}$ with $mu(A)>0$, and any $epsilon>0$, the sets of returns $$ R_{epsilon, A}= big{ninmathbb{N}:mu(Acap T^{-lfloor f(n)rfloor}A)>mu^2(A)-epsilonbig} $$ and $$ R^{(k)}_{A}= big{ ninmathbb{N}: mubig(Acap T^{lfloor f(n)rfloor}Acap T^{lfloor f(n+1)rfloor}Acapcdotscap T^{lfloor f(n+k)rfloor}Abig)>0big} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.
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