In 1975, mathematician Endre Szemerédi proved a conjecture that sparked various lines of research still being explored by mathematicians today. His work led to the question of the largest fraction of a set of numbers that can be used without including a forbidden progression.
For instance, if you have a pool of numbers ranging from 1 to 20, you can write down 16 numbers without creating progressions of five or more numbers. This equates to 80% of the starting pool. However, as the pool size increases, the fraction of numbers that can be used decreases, eventually shrinking to zero as the pool grows.
Mathematicians have made significant progress in understanding progression patterns in sets of numbers. While recent work has provided insights into three-term progressions, avoiding longer progressions remains challenging due to the complex equations they must satisfy. Timothy Gowers developed a theory in the late 1990s to address this issue, earning him the prestigious Fields Medal.
In 2022, graduate student Leng embarked on a mission to understand Gowers’ theory, eventually making groundbreaking progress with the help of Sah and Sawhney. They improved the upper bound on sets without five-term progressions, making the first significant advancement in 23 years. Their work revealed that as the pool size increases, progression-avoiding sets become relatively smaller at an exponentially faster rate.
The method used by the trio to achieve this new bound has sparked excitement among mathematicians. By strengthening an older result by Green, Terence Tao, and Tamar Ziegler, they were able to enhance the understanding of Gowers’ theory. Mathematicians believe that further improvements can be made to this theory, unlocking more insights into progression patterns in sets of numbers.
Sawhney has since completed his Ph.D. and is now a Clay Fellow at Columbia University, while Sah continues his graduate studies at MIT. Their collaboration has impressed many in the mathematical community, showcasing their ability to tackle complex and demanding problems with innovative solutions. The trio’s accomplishments have been lauded as a significant achievement in a field known for its difficulty and intricacy.
