A monad measure space for logarithmic density

TL;DR

This paper develops a measure-theoretic framework using monad measure spaces to analyze sets with positive Banach logarithmic density, establishing new structural theorems about approximate geometric progressions and multiplicative sumsets.

Contribution

It introduces a novel measure space approach employing Loeb measures and multiplicative cuts to prove structural theorems for sets with positive Banach logarithmic density.

Findings

Sets with positive Banach logarithmic density contain approximate geometric progressions.

Positive Banach logarithmic density sets have multiplicatively bounded gaps in their product.

The framework generalizes Jin's sumset theorem to a multiplicative setting.

Abstract

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $A\subseteq \mathbb{N}$ has positive Banach logarithmic density, then $A$ contains an approximate geometric progression of any length. We also prove that if $A,B\subseteq \mathbb{N}$ have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $A\cdot B$ are multiplicatively bounded, a multiplicative version Jin's sumset theorem. The main technical tool is the use of a quotient of a Loeb measure space with respect to a multiplicative cut.