$|P| = N$, then $| \Delta_K(P) | \gtrsim N^{3/4}$, with stronger estimates established by Garibaldi in special cases ([@Gar04]). There is a conversion mechanism to go from Falconer-type results to Erdős-type results that was developed by the second author together with Hoffman ([@HI05]), Laba ([@IL05]), and Rudnev and Uriarte-Tuero ([@IRU14]). It gives estimates for point sets that are fairly spread out. Applying the conversion mechanism to Theorem \[main2\] we get the following corollary:

\[falconertoerdosthm\] Let $K$ be a symmetric convex body in ${\Bbb R}^2$ whose boundary $\partial K$ is $C^\infty$ smooth and has strictly positive curvature. Let $P$ be a set of $N$ points in ${[0,1]}^2$ so that the distance between any two points is $\gtrsim N^{-1/2}$. Then there exists $x \in P$ such that $$\label{erdospinnedours} | \Delta_{K,x}(P)| \gtrapprox N^{\frac{4}{5}}.$$

The main obstacle
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The work on the Falconer problem by Wolff [@W99] and Erdoğan [@Erd05] is based on a framework developed by Mattila ([@Mat85] and [@Mat87]) which connects the original geometric problem to estimates in Fourier analysis. Suppose that $E$ is a compact set with positive $\alpha$-dimensional Hausdorff measure. Then there is a probability measure $\mu$ supported on $E$ with $\mu(B(x,r)) \lesssim r^\alpha$ for every ball $B(x,r)$. The measure 