  Given a level set $E$ of an arbitrary multiplicative function $f$, we
establish, by building on the fundamental work of Frantzikinakis and Host
[13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into
an almost periodic and a pseudo-random parts. Using this structure theorem
together with the technique developed by the authors in [3], we obtain the
following result pertaining to polynomial multiple recurrence.
  Let $E=\{n_1<n_2<\ldots\}$ be a level set of an arbitrary multiplicative
function with positive density. Then the following are equivalent:
  - $E$ is divisible, i.e. the upper density of the set $E\cap u\mathbb{N}$ is
positive for all $u\in\mathbb{N}$;
  - $E$ is an averaging set of polynomial multiple recurrence, i.e. for all
measure preserving systems $(X,\mathcal{B},\mu,T)$, all $A\in\mathcal{B}$ with
$\mu(A)>0$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$,
$i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$
\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N \mu\big(A\cap
T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$
  We also show that if a level set $E$ of a multiplicative function has
positive upper density, then any self-shift $E-r$, $r\in E$, is a set of
averaging polynomial multiple recurrence. This in turn leads to the following
refinement of the polynomial Szemer\'edi theorem (cf. [4]).
  Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$
has positive upper density and let $r\in E$. Then for any set $D\subset
\mathbb{N}$ with positive upper density and any polynomials
$p_i\in\mathbb{Q}[t]$, $i=1,\ldots,\ell$, which satisfy
$p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all
$i\in\{1,\ldots,\ell\}$, there exists $\beta>0$ such that the set $$
\left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n))
\Big)>\beta \,\right\} $$ has positive lower density.
