# A notion of largeness somewhere between $\mathrm{IP}_+$ and $\mathrm{IP}_+^*$

It is a well known fact that a set $$A \subset \mathbb{N}$$ is $$\mathrm{IP}$$ if and only if there exists an idempotent $$p \in \beta \mathbb{N}$$ (i.e., $$p+p=p$$) such that $$A \in p$$. Similarly, $$B \subset \mathbb{N}$$ is $$\mathrm{IP}^*$$ if and only if $$B \in p$$ for each idempotent $$p \in \beta \mathbb{N}$$. I am interested in the following property of a set $$A \subset \mathbb{N}$$:

$$\text{For each idempotent } p \in \beta \mathbb{N} \text{ there are } A' \in p,\ a \in \mathbb{N},\ b \in \mathbb{Z} \text{ s.t. } aA'+b\subset A \tag{\star}.$$

Clearly, any $$\mathrm{IP}_+^*$$ set (that is, a shift of an $$\mathrm{IP}^*$$ set) has the property $$(\star)$$. Also, since a dilation of an $$\mathrm{IP}$$ set is again and $$\mathrm{IP}$$ set, any set with the property $$(\star)$$ is an $$\mathrm{IP}_+$$ set (that is, a shift of an $$\mathrm{IP}$$ set). Slightly more generally, taking particular choices of $$p$$ one can shows that $$(\star)$$ implies $$\text{central}_+$$, positive upper asymptotic density and presumably a few other properties.

Has this property been previously studied? Does it have a simpler combinatorial characterisation?

• Where might I learn more about the "well known fact" that you start your question with? – Gro-Tsen Feb 19 '19 at 14:53
• Gro-Tsen: Hindman and Strauss's book Algebra in the Stone-Čech Compactification is a comprehensive reference with detailed proofs of this and related facts. It's a fun read. – John Griesmer Feb 23 '19 at 2:19