independently by Xiumin Du and Ruixiang Zhang (personal communication). It may be of independent interest.

Here is an outline of the paper. In Section \[sec:outline\], we set up our framework (defining ${\mu_{1, good}}$ precisely) and outline the main estimates. At that point, we will be able to make some further comments about the proofs of the two main propositions. In Section \[sec:mainest1\], we prove Proposition \[mainest1\]. In Section \[sec:RS\], we state and prove a refinement of the decoupling theorem. At that point, we will give some more context about this result. Section 4 does not depend on any previous sections. In Section \[sec:mainest2\], we prove Proposition \[mainest2\] by combining Liu’s framework with our decoupling tools. This will finish the proof of Theorem \[main\]. In Section \[sec:traintracks\], we present in detail the train track example that we introduced above. In Section \[sec:genmetric\], we adapt our arguments to general metrics $\| \cdot \|_K$, proving Theorem \[main2\]. In Section \[sec:erdos\], we prove Corollary \[falconertoerdosthm\].

[**Acknowledgements.**]{} The first author is supported by a Simons Investigator grant. The second author is supported in part by the NSA Grant H98230-15-0319. The third author is supported by NSF-DMS \#1764454.

Setup and outline of the main estimates {#sec:outline}
