Katz–Tao discretized sum-product conjecture

For $0<\sigma<1$, define a $(\delta,\sigma)_1$-set to be a $\delta$-separated subset $A\subset[1,2]$ satisfying $|A|\sim\delta^{-\sigma}$ and $|A\cap I|\lesssim |I|^{\sigma}|A|$ for every interval $I$. Let $\mathcal{N}(B,\delta)$ denote the maximal cardinality of a $\delta$-separated subset of $B$. Katz–Tao discretized sum-product conjecture. There exists $c=c(\sigma)>0$ such that every $(\delta,\sigma)_1$-set $A\subset[1,2]$ satisfies

$$\max\{\mathcal{N}(A+A,\delta),\mathcal{N}(AA,\delta)\}\gtrsim \delta^{-c}|A|.$$

This is a discretized analogue of the Erdős–Szemerédi problem, with covering numbers replaced here by maximal separated-set cardinalities. The source attributes the question to Katz and Tao; its resolution status is not given in the supplied text.