A structure theorem for level sets of multiplicative functions and applications

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 $E=\{n_1<n_2<\dots \}$ 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 $\mathrm{\ell }\ge 1$ and all polynomials $p_i\in \mathbb{Z}[x]$, $i=1,\dots ,\mathrm{\ell }$, with $p_i(0)=0$ we have $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{j=1}^N\mu (A\cap T^{-p_1(n_j)}A\cap \dots \cap T^{-p_{\mathrm{\ell }}(n_j)}A)>0.$$ 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}$$withpositiveupperdensityandanypolynomials$p_i\in$\mathbb{Q}$[t]$,$i=1,\ldots,\ell$,whichsatisfy$p_i($\mathbb{Z}$)\subset$\mathbb{Z}$$and$p_i(0)=0$forall$i\in{1,\ldots,\ell}$,thereexists$\beta>0$suchthattheset$$ \left{,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>\beta ,\right} $$ has positive lower density.