Let ${\mathbb N}$ denote the set of positive integers, let $A,B\subseteq \mathbb{N}$. For $n\in\mathbb{N}$ we set $n+A:=\{n+a: a\in A\}$. We say that $A$ *is infinitely often in* $B$ if the set $$\big\{n\in\mathbb{N}:(n+A)\subseteq B\big\}$$ is infinite.
Moreover, for any $S\subseteq {\mathbb N}$ we set $\mu(S) = \lim \inf_{n\to\infty}\frac{|S\cap\{1,\ldots,n\}|}{n}.$

**Question.** If $B\subseteq \mathbb{N}$ and $\mu(B) > 0$, does the following statement always hold?

(S): For any $n\in \mathbb{N}$ there is $A\subseteq \mathbb{N}$ with $|A|=n$ and $A$ is infinitely often in $B$.

If not, is there a positive $r\in\mathbb{R}$ with $r<1$ such that whenever $B\subseteq \mathbb{N}$ has the property that $\mu(B)\geq r$ then (S) holds? (I will accept answers of the first question, but if it is negative, I would be interested in remarks concerning the latter question.)