Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that $$ G = \bigsqcup_{b\in B} bA.$$ In other words, $G$ is equal to the product set $BA$ and the left-translates $\{ bA : b \in B\}$ are disjoint.
Further, let's call $A$ a $\delta$-approximate tiling of $G$ if there exists a $B$ where again, the left-translates are disjoint, but instead of $BA = G$, one has $$|BA| \geqslant (1-\delta)|G|.$$ Here $0 \leqslant \delta < 1$ is a parameter that doesn't depend on $A$ or $G$, and so $BA$ is a positive proportion of $G$.
For our application, we need to understand in what families of groups $G$, one has a $\delta$-approximate tiling of size $|A| \approx \log_2 |G|$ -- say $|A| = \log_2 |G| + O(\log\log|G|)$ where the implicit constant can depend on the family but not on the individual group.
To give some examples, in the family of cyclic groups $G = \mathbb{Z}/N\mathbb{Z}$, it's fairly easy to see that one gets a $(1-\epsilon)$-approximate tiling of the desired size by taking the set $A = \{1, \cdots, k\}$ where $k$ is chosen so that $N = 2^k k^{O(1)}$; one can take $B$ to be multiples of $k$. Another example is the family of vector spaces $G = \mathbb{F}_2^d$. In this case, one can choose $A$ to be a subspace of the appropriate size and $B$ to be a complete collection of coset representatives, which one can do since $\mathbb{F}_2^d$ has many, many subspaces as $d \to \infty$.
Here are two concrete questions:
Is there a family of groups such that there is no approximate tiling of size $\log_2 |G|$ at all?
Are there alternative names for the concept of tilings or approximate tilings in the literature? I picked this nomenclature in analogy with the concept of tiling in abelian harmonic analysis (say over finite abelian groups), but I suspect someone may have looked at a similar concept for nonabelian groups under a different name.
