The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ is a (bounded) function ($R < \infty$ is fixed), then there are (at most) countably many points $\{a_j\}_j$ such that $\{B(a_j, r(a_j))\}_j$ is a cover of $A$ with multiplicity at most $N(d)$ (i.e. for any $j$, $B(a_j, r_j)$ has nonempty intersection with at most $N(d) - 1$ balls $B(a_k, r_k)$, $k \neq j$).

I should underscore the fact that the the multiplicity $N(d)$ depends only on $d$ and on none of: the 'ball size function' $r$, the bound $R$, or the set $A$.

I am interested in a generalization of this fact to manifolds, replacing $\mathbb{R}^d$ with a (finite dimensional, smooth, Riemannian) manifold $M$ and Euclidean balls with geodesic balls. Unfortunately, it seems that the BCL is generally false in this setting, due to a result of Chi: indeed, according to that paper, if the BCL holds on a Hadamard manifold (i.e. everywhere nonpositive sectional curvature), then that manifold is $\mathbb{R}^d$ for some $d$.

So, one must compromise to obtain a positive result. I think that it suffices to constrain the maximum ball size $R$: given a manifold $M$, the BCL holds for ball size functions $r \leq R$, where $R$ depends on $M$. For e.g., one can apply the Nash embedding theorem and the classical BCL, and this implies the following: for any compact $M$, there is are numbers $N = N(M)$ and $R = R(M) > 0$ such that for any $A \subset M$ and any $r : A \to (0,R]$, there is an at-most countable cover $B(a_j, r(a_j)$ of $A$ by geodesic balls with multiplicity $\leq N$.

**Question:** Is there any way to tell what $R$ should be by looking at only local features of the manifold?

The constraint $R$ comes from how 'folded up' the Nash/Whitney embedding is in $\mathbb{R}^d$, and I have no clue how to control it. So, this result is highly unsatisfying: for small enough $R$, the BCL is a local result (isn't it?), and so the multiplicity shouldn't have anything at all to do with the global 'folding up' of an embedding in $\mathbb{R}^d$. Instead, I suspect that one ought to be able to cook up a constraint on $R$ that depends only on some local feature, like the sectional or Ricci curvatures. Has anyone done this or know of prior work in this direction?