# 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 '16**Published on: **01 Nov '16**Published in: **Monatshefte für Mathematik**

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#### 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
\(A,B\subseteq \mathbb {N}\)
\(A,B\subseteq \mathbb {N}\)
\(A\cdot B\)
\(A\cdot B\)