Erdős–Szemerédi theorem

In arithmetic combinatorics, the Erdős–Szemerédi theorem, proven by Paul Erdős and Endre Szemerédi in 1983,[1] 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 ${\displaystyle \varepsilon }$ such that

${\displaystyle \max(|A+A|,|A\cdot A|)\geq c|A|^{1+\varepsilon }}$

whenever A is a finite non-empty set of real numbers of cardinality |A|, where ${\displaystyle A+A=\{a+b:a,b\in A\}}$ is the sum-set of A with itself, and ${\displaystyle A\cdot A=\{ab:a,b\in A\}}$.

It is possible for A + A to be of comparable size to A if A is an arithmetic progression, and it is possible for A · A to be of comparable size to A if A is a geometric progression. The Erdős–Szemerédi theorem can thus be viewed as an assertion that it is not possible for a large set to behave like an arithmetic progression and as a geometric progression simultaneously. It can also be viewed as an assertion that the real line does not contain any set resembling a finite subring or finite subfield; it is the first example of what is now known as the sum-product phenomenon, which is now known to hold in a wide variety of rings and fields, including finite fields.[2]

It was conjectured by Erdős and Szemerédi that one can take ${\displaystyle \varepsilon }$ arbitrarily close to 1. The best result in this direction currently is by George Shakan,[3] who showed that one can take ${\displaystyle \varepsilon }$ arbitrarily close to ${\displaystyle 1/3+5/5277}$. Previously, Misha Rudnev, Ilya Shkredov, and Sophie Stevens had shown that one can take ${\displaystyle \varepsilon }$ arbitrarily close to ${\displaystyle 1/3+1/1509}$,[4] improving an earlier result by József Solymosi,[5] who had shown that one can take it arbitrarily close to ${\displaystyle 1/3}$.