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Published November 2016 | Submitted
Journal Article Open

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.

Additional Information

© 2016 Springer-Verlag Wien. Received: 02 April 2015; Accepted: 07 September 2016; First Online: 14 September 2016. The authors were supported in part by the American Institute of Mathematics through its SQuaREs program. I. Goldbring was partially supported by NSF CAREER Grant DMS-1349399. M. Lupini was supported by the York University Susan Mann Dissertation Scholarship. K. Mahlburg was supported by NSF Grant DMS-1201435.

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August 22, 2023
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October 23, 2023