Additive combinatorics and multiplicative functions

This project will advance fundamental mathematical knowledge by developing new tools to understand the deep connections between addition and multiplication within number systems. It bridges two major areas of pure mathematics: additive combinatorics, which studies patterns formed by adding numbers, and multiplicative number theory, which focuses on properties related to multiplying numbers, especially prime numbers. The PI will create novel techniques to solve complex problems where these additive and multiplicative structures interact. Findings of the project will benefit fields beyond mathematics, particularly cybersecurity, where improved understanding of number-theoretic functions could lead to stronger encryption methods protecting digital information. The combinatorial methods developed will also have the potential to enhance the reliability of data transmission and storage systems. Additionally, the project will train undergraduate and graduate students and contribute to educational outreach through math competitions and K-12 programs to inspire future scientists. The project will tackle challenging open questions involving prime numbers, smooth numbers, and general multiplicative functions. More specifically, the PI will develop new methods in additive combinatorics that can be applied to questions involving primes, such as finding narrow progressions in primes and finding solutions to linear equations in subsets of primes. The PI will analyze the behavior of multiplicative functions in various additive contexts, such as obtaining Gowers norm estimates for multiplicative functions in short intervals and in arithmetic progressions. The project aims to advance techniques in additive combinatorics, especially those based on higher order Fourier analysis and the quantitative theory of inverse sumset theorems, to tackle problems involving both additive and multiplicative structures. By studying how multiplicative functions behave in additive structures, the project seeks to uncover deeper insights into their distribution properties and connections to prime numbers and other key objects in number theory. The project is also expected to forge new connections across mathematics and theoretical computer science, stimulating progress in these fields. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.