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