Erdős–Szemerédi theorem

In arithmetic combinatorics, the Erdős–Szemerédi theorem, proven by Paul Erdős and Endre Szemerédi in 1983, states that, for every finite set of real numbers, either the pairwise sums or the pairwise products of the numbers in the set form a significantly larger set. More precisely, it asserts the existence of positive constants c and $\varepsilon$ such that
$\max(|A+A|,|A\cdot A|)\geq c|A|^{1+\varepsilon }$ whenever A is a finite non-empty set of real numbers of cardinality |A|, where $A+A=\{a+b:a,b\in A\}$ is the sum-set of A with itself, and $A\cdot A=\{ab:a,b\in A\}$ .
It was conjectured by Erdős and Szemerédi that one can take $\varepsilon$ arbitrarily close to 1. The best result in this direction currently is by George Shakan, who showed that one can take $\varepsilon$ arbitrarily close to $1/3+5/5277$ . Previously, Misha Rudnev, Ilya Shkredov, and Sophie Stevens had shown that one can take $\varepsilon$ arbitrarily close to $1/3+1/1509$ , improving an earlier result by József Solymosi, who had shown that one can take it arbitrarily close to $1/3$ .