Does Szemerédi's theorem hold for sets with positive upper Banach density?

Question

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?

Answer 1

Yes. As Martin says, this is often how the theorem is stated. It also follows immediately from the also common finitary form:

For all $\delta>0$ and $k\geq 1$, if $N$ is large enough depending on $\delta$ and $k$, $P$ is an arithmetic progression of length $N$, and $A\subseteq P$ has size $\lvert A\rvert\geq \delta N$, then $A$ contains a non-trivial arithmetic progression of length $k$.

(This is e.g. equivalent to the finitary form stated in the Wikipedia article where $P=\{1,\ldots,N\}$, since the property of containing an arithmetic progression is invariant under dilations and translations.)