2011/05/27 by Tim Austin, Austin, Tim
Computer Science · Mathematics · #05D99 #28D05 #37A45 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.CO #math.DS #msc:05D99 #msc:28D05 #msc:37A45
paper · pdf · doi:10.48550/arxiv.1105.5611
33 pages. [TDA Aug 26, 2011:] Replaced with minor corrections
openalex publication_date 2011/05/27 · arxiv created 2013/07/21 · arxiv updated 2013/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We exhibit proofs of two ergodic-theoretic results in the study of multiple recurrence using an analog of the density-increment argument of Roth and Gowers: Furstenberg's Multiple Recurrence Theorem (which implies Szemerédi's Theorem), and a two-dimensional special case of Furstenberg and Katznelson's multidimensional version of this theorem. The second of these requires also an analog of some recent finitary work by Shkredov. Many proofs of these multiple recurrence theorems are now known, but our main goal is to shed some further light on the heuristic correspondence principle that has grown up between the ergodic-theoretic and combinatorial aspects of multiple recurrence and Szemerédi's Theorem. Focusing on the density-increment strategy highlights several close points of connection between these settings.