Question

Following Rosenblatt and Yang, I say that a subset $A$ of $\mathbb Z$ has a unique mean value if for all invariant means $\lambda_1,\lambda_2$ on $\mathbb Z$, one has $\lambda_1(A)=\lambda_2(A)$.

Notice that the set of subsets having a unique mean value has the same cardinality as the set of subsets non-having a unique mean value, thanks to Andreas Thom's answer to http://mathoverflow.net/questions/65325/intrinsically-measurable-subsets-of-amenable-semigroups.

Nevertheless, roughly speaking, it should be clear that most subsets of $\mathbb Z$ should not have a unique mean value. Indeed, to a have a unique mean value one needs a very particular structure, as almost-periodicity.

Question: Is there any probabilistic way to formalize the intuition that generic subsets of $\mathbb Z$ does not have a unique mean value?

Thank you in advance,

Valerio

Answer 1

If a set $A$ has arbitrarily large gaps, meaning that there are intervals $I_k$ of lengths tending to infinity that are disjoint from $A$, then there is a mean $\lambda$ such that $\lambda(A)=0$: take $\lambda(X)$ be a (Banach) limit of $|X \cap I_k| / |I_k|$. Similarly, if $A$ contains arbitrarily long intervals, then there is a mean $\mu$ such that $\mu(A)=1$.

So, for example, the set of subsets not having a unique mean is a dense $G_\delta$ set inside $2^ \mathbb{Z}$. Also, if you randomize a set and the events that $n$ belongs to the set are independent (and don't have probability 0 or 1), then, with probability 1, the set you get will not have a unique mean.