The discretized ring conjecture of Katz and Tao

Let $n$, $\alpha$, $\varepsilon$, $C$, and $\delta$ be parameters chosen in that order. A set $E\subset\mathbb{R}^n$ is a $(\delta,\alpha)_n$-set if it is a union of balls of radius $\delta$ and, for every ball $B$ of radius $r$,

$$|E\cap B|\leq C\delta^{n-\varepsilon}(r/\delta)^\alpha.$$

Katz–Tao's discretized ring conjecture. There exists an absolute constant $c_1$ such that, if $\varepsilon>0$ is sufficiently small and $C$ is sufficiently large, then for all sufficiently small $\delta>0$, every $(\delta,1/2)_1$-set $E\subset[1,2]$ satisfying $|E|\geq\delta^{1/2+\varepsilon}$ obeys

$$|E+E|+|E.E|\geq\delta^{-c_1}|E|.$$

This is a discretized form of the Erdős–Szemerédi sum-product conjecture and is connected to the Falconer distance problem and Furstenberg-set dimension estimates. The source presents it as Katz and Tao's discretized ring conjecture; its resolution status is not specified here.