the bad tubes going through $x$. Recall that a tube $T$ is bad if its $\mu_2$ measure is too large. In general a point $x$ could lie in many bad tubes, and we need to know that the total $\mu_1$ measure of all these bad tubes is small (for most $x \in E_2$). This follows from Orponen’s radial projection theorem from [@O17b]. This theorem plays an important role in Keleti and Shmerkin’s work on the Falconer problem [@KS18], which is where we learned about it.

To discuss Proposition \[mainest2\], we first describe the framework from [@Liu18]. Let $\sigma_t$ denote the normalized arc length measure on the circle of radius $t$ (so the total measure is 1). In [@Liu18], Liu proved the following remarkable identity: for any function $f$,

$$\int_0^\infty | f * \sigma_t (x)|^2 t dt = \int_0^\infty |f * \hat \sigma_r(x)|^2 r dr.$$

It follows from this identity that

$$\int_{E_2} \| d^x_* ({\mu_{1, good}}) \|_{L^2}^2 d \mu_2(x) \lesssim \int_0^\infty \left( \int_{E_2} |{\mu_{1, good}}* \hat \sigma_r|^2 d\mu_2 (x) \right) r dr.$$

Now the Fourier transform of ${\mu_{1, good}}* \hat \sigma_r$ is supported on the circle of radius $r$, and studying such functions is the subject of restriction theory. We can 