Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/study before in any article whether left and right syndeticity are equal or not in nilpotent groups?

To elaborate the question here I recall the notion of left and right syndeticity [Definition 4.38] from the book 'Algebra in the Stone-Cech compactification' by Hindman and Strauss.

Definition: Let $S$ be a semigroup. Let $A\subseteq S$.

1. The set $A$ is right syndetic if there exists some $G\in\mathcal{P}_f(S)$ such that $S=\bigcup_{t\in G}t^{-1}A$.
2. The set $A$ is left syndetic if there exists some $G\in\mathcal{P}_f(S)$ such that $S=\bigcup_{t\in G}At^{-1}$.

Note that $\mathcal{P}_f(S)$ stands for the collection of all nonempty finite subsets of $S$.

Thanks in advance for any help or suggestion.

$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/study before in any article whether left and right syndeticity are equal or not in nilpotent groups?

To elaborate the question here I recall the notion of left and right syndeticity [Definition 4.38] from the book 'Algebra in the Stone-Cech compactification' by Hindman and Strauss.

Definition: Let $S$ be a semigroup. Let $A\subseteq S$.

1. The set $A$ is right syndetic if there exists some $G\in\Pf(S)$ such that $S=\bigcup_{t\in G}t^{-1}A$.
2. The set $A$ is left syndetic if there exists some $G\in\Pf(S)$ such that $S=\bigcup_{t\in G}At^{-1}$.

Note that $\Pf(S)$ stands for the collection of all nonempty finite subsets of $S$.

Thanks in advance for any help or suggestion.