The Falconer distance conjecture

Theorem 1.1: for every integer $d\ge2$ and every compact $E\subset\mathbb R^d$ with $\dim_H E>d/2$, $\mathcal L^1(\Delta(E))>0$. No regularity beyond the strict dimension bound is assumed. The paper does NOT treat the endpoint $\dim_H E=d/2$, does NOT prove a pinned version (some $y$ with $\mathcal L^1(\Delta_y(E))>0$) at the threshold $d/2$. It states that it uses no decoupling theorem and no earlier distance theorem at an improved threshold; it does use the Orponen-Shmerkin incidence theorem.

Theorem 1.1: for every integer $d\ge2$ and every compact $E\subset\mathbb R^d$ with $\dim_H E>d/2$, $\mathcal L^1(\Delta(E))>0$. No regularity beyond the strict dimension bound is assumed. The paper does NOT treat the endpoint $\dim_H E=d/2$, does NOT prove a pinned version (some $y$ with $\mathcal L^1(\Delta_y(E))>0$) at the threshold $d/2$. It states that it uses no decoupling theorem and no earlier distance theorem at an improved threshold; it does use the Orponen-Shmerkin incidence theorem.