# A monad measure space for logarithmic density

Research paper by Mauro Di Nasso, Isaac Goldbring; Renling Jin; Steven Leth; Martino Lupini; Karl Mahlburg

Indexed on: 08 Oct '16Published on: 01 Nov '16Published in: Monatshefte für Mathematik

#### Abstract

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.AbstractWe 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. $$A\subseteq \mathbb {N}$$ $$A\subseteq \mathbb {N}$$A $$A,B\subseteq \mathbb {N}$$ $$A,B\subseteq \mathbb {N}$$ $$A\cdot B$$ $$A\cdot B$$