2025/03/19 by Nikos Frantzikinakis, Frantzikinakis, Nikos · 1 citation
Computer Science · Mathematics · #11B30 #11N37 #37A15 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary: 37A44 #Secondary:05D10 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2503.15175
openalex publication_date 2025/03/19 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
Motivated by partition regularity problems of homogeneous quadratic equations, we prove multiple recurrence and convergence results for multiplicative measure preserving actions with iterates given by rational sequences involving polynomials that factor into products of linear forms in two variables. We focus mainly on actions that are finitely generated, and the key tool in our analysis is a decomposition result for any bounded measurable function into a sum of two components, one that mimics concentration properties of pretentious multiplicative functions and another that mimics vanishing properties of aperiodic multiplicative functions. Crucial to part of our arguments are some new seminorms that are defined by a mixture of addition and multiplication of the iterates of the action, and we prove an inverse theorem that explicitly characterizes the factor of the system on which these seminorms vanish.