Let $G$ be a at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.

It is well known that if $G$ is a finite group then this action admits a fundamental domain.

Call a set $U\subset X$ an $(\varepsilon, N)$-fundamentalish domain iff there exist $N$ elements $g_1, \ldots, g_N$ of $G$ such that the sets $g_i(U)$ are pairwise disjoint and the measure of their union is at least $1-\varepsilon$.

Question 1 If $G$ is an infinte group, N_0 is a natural number, $\varepsilon_0$ is a positive real number, does there exist an $(\varepsilon, N)$-fundamentalish domain with $\varepsilon<\varepsilon_0$ and $N>N_0$?

Question 2 Same as above, but assume there is an element of infinite order in $G$.

Question 2 boils down to $G=\mathbb Z$. I'm particularly interested in Bernoulli shifts.

For example when the action is profinite and "transitive on each level", then clearly answer to Question 1 is positive: there exist $(0,N)$-fundamentalish domains for arbitrary large $N$.