A different version of Besicovitch Covering Theorem involving balls of half radius

Question

I am trying a find a reference to/proof of the following result:

Let $(M, g)$ be a compact Riemannian manifold. Then there is $b$ so that the following holds: for any $r>0$, there is a covering $\mathscr U$ of $M$ by open geodesic balls of radius $r$ so that (i) every member of $\mathscr U$ intersects at most $b$ other members of $\mathscr U$, and (ii) the collection $$ \mathscr U_{1/2} = \{ B(a, r/2) : B(a, r) \in \mathscr U\}$$ still covers $M$.

Without condition (ii), it is a consequence of Besicovitch Covering Theorem. However, (ii) cannot holds in the general form (e.g. $r$ is not constant) as was shown in the MO post Stronger version of Besicovitch covering theorem.

I am guessing the fact that the radius is constant (or bounded by below by some positive constant) would be sufficient. The above theorem is used quite frequently: in p.16, start of proof of Proposition 4.3 in Sacks and Uhlenbeck - The existence of minimal immersions of 2-spheres, the last paragraph in p.390 of Choi and Schoen - The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, and a more recent occurrence: p.23 of Chen and Warren - Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume, where they cited the Besicovitch Covering Theorem.

Answer 1

Are you allowing $b$ to depend on the manifold (as it appears to me from your statement)? In that case, Besicovitch is overkill, and this statement holds in much more generality than Besicovitch. One only needs to know that a compact Riemannian manifold is a doubling metric space. Doubling has many equivalent definitions, but here you can think of it as: any ball of radius $s$ contains at most $C$ $s/2$-separated points, for some fixed constant $C$ independent of $s$.

With that, let $N$ be a maximal $\frac{r}{2}$-separated set in $M$, i.e., a set of points such that any two have distance $\geq \frac{r}{2}$ and such that every point in $M$ is distance $<\frac{r}{2}$ away from one of them. Let $\mathcal{U}$ be the balls of radius $r$ centered on $N$. By construction $\mathcal{U}$ satisfies (ii) immediately. Property (i) follows from the doubling condition: The centers of all balls in $\mathcal{U}$ that touch a fixed ball from $\mathcal{U}$ form an $\frac{r}{2}$-separated set inside a ball of radius $2r$, and the cardinality of such a set is controlled by the doubling constant.

To see that every compact Riemannian manifold $M$ is doubling is easy. Cover $M$ by finitely many charts that are bi-Lipschitz to subsets of $\mathbb{R}^d$, and the doubling constant can then be estimated based on the number of charts, the bi-Lipschitz constants, and the doubling constant of $\mathbb{R}^d$. More refined estimates based on curvature are possible too; it depends what you need.