Topic: Arithmetic progressions

Raghu Meka and Zander Kelley examine how large a subset of {1, …, N} must be to contain a three-term arithmetic progression. Writing its size as at least N/C, the talk places the problem between Roth’s classical guarantee at C approximately log log N and Behrend’s progression-free construction at an exponential scale in the square root of log N. It recalls Bloom and Sisask’s 2020 improvement to C = (log N)^(1+c), for some c > 0, before giving an in-depth account of the Kelley–Meka proof reaching C approximately 2^((log N)^0.09), moving closer to Behrend’s scale.

Thomas Bloom surveys quantitative bounds for sets of integers without three-term arithmetic progressions, beginning with Roth’s zero-density theorem. Work with Olof Sisask establishes that a set with divergent reciprocal sum must contain a three-term progression, resolving the first nontrivial case of Erdős’s conjecture. The proof combines harmonic analysis with elementary combinatorics. The second part develops a structural theorem for additively non-smoothing sets: sets whose first sumset grows, while subsequent addition reveals no additional structure. Building on Bateman and Katz’s cap-se