We say that a set of natural numbers $A\subseteq \omega$ has *positive upper density* if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$
Szeméredi's theorem states that every $A\subseteq \omega$ having positive upper density contains arithmetic sequences of arbitrary (finite) length.

For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ equals the difference of $a$ and $y$ if $a\geq y$, and $0$ otherwise. The *upper Banach density* is defined by $$d_{uB}(A) = \lim_{n\to\infty}\big(\sup
\{\frac{|(A-y)\cap n|}{n+1}: y\in\omega\}\big).$$
It is easy to see that $d_{uB}(A) \geq d_{u}(A)$ for all $A\subseteq \omega$.

**Question.** If $A\subseteq \omega$ has the property that $d_{uB}(A)>0$, does $A$ contain arithmetic sequences of arbitrary finite length?