Suppose that $X$ is a topological space, and $T$ is a topological group which continuously acts on $X$ on the right. We call the pair $(X,T)$ a (right) transformation group.
We know that $(X,\mathbb R)$ is a topological group. Fix $x\in X$ and let $U$ be an open neighborhood of $x$. Let $A_x(U)$ be the set of all $t\in \mathbb R$ such that $x.t\in U$. We call $x$ an almost periodic point if for every neighborhood $U$ of $x$ there is a compact set $K_x(U)$ such that $\mathbb R=K_x(U)+A_x(U)$. Is it true that this defining condition requires $A_x(U)$ to be relatively dense?
I solved this already trivial question: Let $B$ be an open ball centered at a fixed $t\in \mathbb R$ of radius $\sup \{|x|: x\in K_x(U)\}$. Write $t$ as the sum $a+k$ with $a\in A_x(U), k\in K_x(U)$. then $a\in B$ and $x.a\in U$. That is, $A_x(U)$ is relatively dense.
It is not necessarily dense
For trivial example:
Let $T$ be any compact group. If $T$ acts trivially, $x\in X$ any point. Then $A_x(U)$ could be empty if $x\not\in U$.
Even if $T$ acts transitively on $X$ the answer is no: For instance take $T$ to be any compact group and $X=T$ (say $X=T=\mathbb{R}/\mathbb{Z}$).
Then for any $x\in X$ and $U\subseteq X$. $A_x(U) = -x+U\subseteq T$. Clearly there exists a compact set $K_x(U)$ such that $T=K_x(U)+A_x(U)$ but $A_x(U)$ is not dense.