Ramsey properties of subsets of $\mathbb{N}$

TL;DR

This paper explores the ergodic properties of WM sets, a class of subsets of natural numbers characterized by weakly mixing dynamical systems, and characterizes which linear Diophantine systems are solvable within all such sets.

Contribution

It provides a complete characterization of linear Diophantine systems solvable in all WM sets and investigates solvability of certain non-linear systems within these sets.

Findings

Complete characterization of linear Diophantine systems solvable in WM sets

Identification of non-linear systems solvable within WM sets

WM sets exhibit algebraic pattern occurrence related to solvability of equations

Abstract

We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain "algebraic patterns" in every subset in the family. By "algebraic pattern" we mean a set of solutions of a system of diophantine equations. In this work we investigate a concrete family of subsets - WM sets. These sets are characterized by the property that the dynamical systems associated to such sets are "weakly mixing", and as such they represent a broad family of randomly constructed subsets of (\mathbb{N}). We find that certain systems of equations are solvable within every WM set, and our subject is to learn which systems have this property. We give a complete characterization of linear diophantine systems which are solvable within every WM set. In addition we study some non-linear equations and systems of equations with regard to the question of solvability within every WM set.