This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for results that state the following: from the given collection, we can select a subcollection $B_1, B_2, ....,B_k, k \leq m$ (relabelled) such that $\cup_{i = 1}^mB_i \subset \cup_{j = 1}^k rB_j$, with $r > 1$ being very close to $1$, and such that the new subcollection has as little intersection between $B_1, B_2,...B_m$ as possible. Of course, phrases "very close to $1$" and "as little intersection" are very vague. I am looking for references which have quantitative results of this type.
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$\begingroup$ Is the corresponding version of the Besikovich covering lemma good enough for your purposes? (the corresponding nearly disjointness property is that every point in the space is contained in at most $N(r,n)$ balls $rB_j$ or, alternatively, the covering family can be partitioned into $N(r,n)$ subfamilies consisting of pairwise disjoint balls) $\endgroup$– fedjaNov 8, 2015 at 21:56
1 Answer
A possible answer in the spirit of your question is the following result of Chanillo and Muckenhoupt: given a number $\delta \in (0, 1/2)$ and a finite collection of balls $B_\alpha$ in $\mathbb{R}^n$, we can find a subcollection $B_1, ..., B_N$ such that $$\bigcup_{\alpha} B_\alpha \subset \bigcup^N_{i = 1}(1 + \delta)B_i,$$ and $$\Sigma_{i = 1}^N \chi_{B_i}(x) \leq 4^n\delta^{-n}.$$ As you can see, the second condition captures "minimal intersection" quantitatively. The proof is pretty nice and elementary, see Lemma 3 of this paper.
Additional comment: The main aim of the paper cited is to improve the famous Donnelly-Fefferman estimates of the BMO norm of $\text{log}(|u|)$, where $u$ is an eigenfunction of the Laplacian $-\Delta$ on a compact Riemannian manifold. Covering lemmas of this type are somewhat common in harmonic analysis though.