A monad measure space for logarithmic density
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 \(A,B\subseteq \mathbb {N}\) \(A,B\subseteq \mathbb {N}\) \(A\cdot B\) \(A\cdot B\)
