Harmonic progressions in additive combinatorics: van der Waerden, Szemerédi, and Green–Tao analogs

This paper studies counting, coloring, and density results on harmonic progressions. We first estimate the number of three-term harmonic progressions. Then, we prove a van der Waerden-type theorem for harmonic progressions, present a Sárközy-type result, and give a height analog of Szemerédi's theorem. Moreover, we give a proof for the harmonic analog of Szemerédi's theorem for multiplicatively closed sets, and we prove the shifted versions of the Green–Tao and Szemerédi theorems for harmonic progressions.