Let $T$ be a topological semigroup. A subset $A$ of $T$ is called syndetic if there is compact subset $K$ of $T$ such that $K^{-1}A= T$. This means that for every $t\in T$, $Kt\cap A\neq \emptyset$. Also a subset $R$ of $T$ is called a thick set if for every compact subset $K$ of $T$, there is $t\in T$ such that $Kt\subseteq R$. I interest to study relation between these notions.
Let $A, R$ be syndetic and thick set for $T$, respectively. I know that $A\cap R\neq \emptyset$.
Q1. Is it true that there is $t\in R$ such that $A\cap t^{-1} A\neq \emptyset$?
Also if for every compact set $K\subseteq T$, we have $S\cap (\bigcap_{k\in K} k^{-1}S)\neq \emptyset$, then $S$ is thick set.
Q2.What can say about the converse of inclusion?
Please help me to know it.