# Solvability of linear equations within weak mixing sets

## TL;DR

This paper introduces WM sets, a new class of 'random' subsets of natural numbers, and establishes conditions for solving linear equations within these sets, including normal and partition-regular systems.

## Contribution

It defines WM sets, explores their properties, and provides necessary and sufficient conditions for linear equation solvability within these sets, extending previous results to a broader class.

## Key findings

- Solvability conditions for linear systems in WM sets
- WM sets include normal sets as a subset
- Partition-regular systems are solvable in any WM set

## Abstract

We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set.