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?


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