We study the limiting behavior of multiple ergodic averages involving
sequences of integers that satisfy some regularity conditions and have
polynomial growth. We show that for "typical" choices of Hardy field functions
$a(t)$ with polynomial growth, the averages $\frac{1}{N}\sum_{n=1}^N
f_1(T^{[a(n)]}x)\cdot...\cdot f_\ell(T^{\ell [a(n)]}x)$ converge in the mean
and we determine their limit. For example, this is the case if $a(t)=t^{3/2},
t\log{t},$ or $t^2+(\log{t})^2$. Furthermore, if $\{a_1(t),...,a_\ell(t)\}$ is
a "typical" family of logarithmico-exponential functions of polynomial growth,
then for every ergodic system, the averages $\frac{1}{N}\sum_{n=1}^N
f_1(T^{[a_1(n)]}x)\cdot...\cdot f_\ell(T^{[a_\ell(n)]}x)$ converge in the mean
to the product of the integrals of the corresponding functions. For example,
this is the case if the functions $a_i(t)$ are given by different positive
fractional powers of $t$. We deduce several results in combinatorics. We show
that if $a(t)$ is a non-polynomial Hardy field function with polynomial growth,
then every set of integers with positive upper density contains arithmetic
progressions of the form $\{m,m+[a(n)],...,m+\ell[a(n)]\}$. Under suitable
assumptions we get a related resultconcerning patterns of the form $\{m,
m+[a_1(n)],..., m+[a_\ell(n)]\}.$