Elekes–Ruzsa conjecture on product sets of small-doubling sets

Let $d4a0\be\bb Z$ be a finite set. Its sumset is $d4a0+d4a0=a+a':a,a'd4a0$. Here $X\buildrel\ll\over\leq Y$ denotes that $X$ is at most a constant multiple of $Y$.

Elekes–Ruzsa conjecture. If

$$|d4a0+d4a0|\ll|d4a0|,$$

then

$$|d4a0\cdotd4a0|\geq|d4a0|^2(\log|d4a0|)^{1-2\log 2-o(1)}.$$

This conjecture concerns the difficult direction of the sum-product phenomenon for sets with bounded additive doubling. The paper describes it as still open and proves results for arithmetic progressions and related settings.