The size of subset of {1,…,N} without 3-term arithmetic progressions is O(N/(log N)^(1+c)) for some c>0

You need to know: A (nontrivial) 3-term arithmetic progression, big O notation, small o notation.

Background: For integer $N>0$ , let $r_3(N)$ be the cardinality of the largest subset of $\{1, 2, \dots , N\}$ which contains no nontrivial 3-term arithmetic progressions.

The Theorem: On 7th July 2020, Thomas Bloom and Olof Sisask submitted to arxiv a paper in which they proved that $r_3(N) = O\left(\frac{N}{\log^{1+c} N}\right)$ for some absolute constant $c>0$ .

Short context: In 1936, Erdős and Turán conjectured that any set containing a positive proportion of integers must contain a 3-term arithmetic progression (3APs). This is equivalent to $\lim\limits_{N\to\infty}\frac{r_3(N)}{N}=0$ . In 1953, Roth confirmed this conjecture by proving that $r_3(N) = O\left(\frac{N}{\log\log N}\right)$ . Another famous conjecture of Erdős states that if A is a set of positive integers such that $\sum\limits_{n \in A}\frac{1}{n}$ diverges then A contains arithmetic progressions of length k for all k. The $k=3$ case of this conjecture was known to follow from $r_3(N) = o\left(\frac{N}{\log N}\right)$ . In 2011, Sanders came close to this by proving that $r_3(N) = O\left(\frac{N}{\log^{1-o(1)} N}\right)$ . The Theorem finally achieves the bound better than $\frac{N}{\log N}$ , and thus implies the $k=3$ case of the Erdős conjecture. Because there are $O\left(\frac{N}{\log N}\right)$ primes up to N, the Theorem also implies the 1939 Van der Corput theorem that the set of primes contains infinitely many 3APs, as well as this generalisation by Green.