2013/03/18 by Nikos Frantzikinakis, Frantzikinakis, Nikos, Bernard Host +1
Mathematics · #05D10 #11B30 #11N37 #37A45 #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1303.4329
openalex publication_date 2013/03/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove partition regularity for certain equations involving forms in three variables, showing for example that the equations 16x2+9y2=n2 and x2+y2-xy=n2 are partition regular, where n is allowed to vary freely in ℕ. For each such problem we establish a density analogue that can be formulated in ergodic terms as a recurrence property for actions by dilations on a probability space. Our key tool for establishing such recurrence properties is a decomposition result for multiplicative functions which is of independent interest. Roughly speaking, it states that the arbitrary multiplicative function of modulus 1 can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of degree three.