Chi Hoi Yip, Georgia Institute of Technology
Inverse sieve problems
Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod p for various sets of primes p; and then sieve methods are the primary tools for doing so. Motivated by the inverse Goldbach problem, Green–Harper, Helfgott–Venkatesh, Shao, and Walsh have explored the inverse sieve problem: if we let S \subseteq N be a maximal set of integers in this interval where the residue classes mod p occupied by S have some particular pattern for many primesp, what can one say about the structure of the set S beyond just its size? In this talk, I will give a gentle introduction to inverse sieve problems, and present some progress we made when S mod p has rich additive structure for many primes p. In particular, in this setting, we provide several improvements on the larger sieve bound for |S|, parallel to the work of Green--Harper and Shao for improvements on the large sieve. Joint work with Ernie Croot and Junzhe Mao.
MC 5479 or Join on Zoom