  We obtain new results pertaining to convergence and recurrence of multiple
ergodic averages along functions from a Hardy field. Among other things, we
confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and
obtain combinatorial applications which contain, as rather special cases,
several previously known (polynomial and non-polynomial) extensions of
Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10;
BMR17]. One of the novel features of our results, which is not present in
previous work, is that they allow for a mixture of polynomials and
non-polynomial functions. As an illustration, assume
$f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and
$a_{i,j}\in\mathbb{R}$. Then
  $\bullet$ for any measure preserving system $(X,{\mathcal B},\mu,T)$ and
$h_1,\dots,h_k\in L^\infty(X)$, the limit
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots
T^{[f_k(n)]}h_k$$ exists in $L^2$;
  $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$
there are $a,n\in\mathbb{N}$ such that $\{a,\,
a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$.
  We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial
growth, and are such that no linear combination of them is a polynomial, then
for any measure preserving system $(X,{\mathcal B},\mu,T)$ and any
$A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu\Big(A\cap
T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,\mu(A)^{k+1}.$$
