A monad measure space for logarithmic density

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $$A\subseteq \mathbb {N}$$A⊆N has positive Banach logarithmic density, then A contains an approximate geometric progression of any length. We also prove that if $$A,B\subseteq \mathbb {N}$$A,B⊆N have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $$A\cdot B$$A·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.