$\mu$ is called a Frostman measure (cf. [@W03], Proposition 8.2.). Let $d(x,y) = |x-y|$. Mattila considered the pushforward measure $d_*(\mu \times \mu)$. Recall that a pushforward measure is defined by

$$\int_{\mathbb{R}}\psi(t) d_*(\mu \times \mu) := \int_{E \times E} \psi(|x-y|) d \mu(x) d \mu(y).$$

In particular $d_*(\mu \times \mu)$ is a probability measure supported on $\Delta(E)$. Mattila noted that if $ \| d_*(\mu \times \mu) \|_{L^2}^2 = \int d_*(\mu \times \mu)(t)^2 dt$ is finite, then Cauchy-Schwarz forces the Lebesgue measure of $\Delta(E)$ to be positive. Then he described an interesting way to rewrite $\| d_*(\mu \times \mu) \|_{L^2}^2$ in terms of the Fourier transform of $\mu$. The resulting integral is connected to restriction theory, and Wolff used that connection to prove the bound in [@W99], building on earlier work by Bourgain [@B94].

In [@Liu18], Liu used a different framework for the Falconer problem which leads to estimates on pinned distance sets. For any $x$, define $d^x(y) = |x-y|$. He studied the quantity

$$\label{liuquant} \int \| d^x_* (\mu) \|_{L^2}^2 d \mu(x).$$

If this key quantity is finite, then for almost every $x \in E$, $\| d^x_* \mu \|_{L^2}$ is finite, and then a Cauchy-Schwarz argument forces the Lebesgue measure of $\Delta_x(E)$ to 