I have the following combinatorial/discrete analysis problem that arose while I was working on a problem in complex analysis, which in turn came from a problem in time-frequency analysis.
Let $\Lambda\subset\mathbb{C}$ be a given uniformly discrete set. Does there exist an $\epsilon>0$ and a uniformly discrete set $\Lambda'\supset\Lambda$ such that $$ \bigcup_{w\in\Lambda'}B(w,f(\Lambda',w)-\epsilon)=\mathbb{C},$$ where $f(\Lambda',w)=\min\{|w'-w|:w'\in \Lambda'\setminus\{w\}\}$. By $S\subset\mathbb{C}$ uniformly discrete I mean a set such that $\inf\{|w_1-w_2|:w_1,w_2\in S, w_1\neq w_2\}>0$.
This has an easy solution if we assume that $\Lambda$ is constrained to a square grid of positive mesh, which is actually all that I needed for the original problem, but I would be nice to know if the general result is true.