Arithmetic progressions and spectral structure

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-set work, Bloom and Sisask extend this kind of structural analysis to arbitrary groups, including the integers. The talk outlines the elementary proof and its potential applications elsewhere in additive combinatorics.