A monad measure space for logarithmic density

Abstract

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

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