Published November 2016
| Version Submitted
Journal Article
Open
A monad measure space for logarithmic density
Creators
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1.
University of Pisa
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2.
University of Illinois at Chicago
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3.
University of California, Irvine
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4.
College of Charleston
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University of Northern Colorado
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6.
York University
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7.
Fields Institute for Research in Mathematical Sciences
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8.
California Institute of Technology
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9.
Louisiana State University
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.Attached Files
Submitted - 1503.03810v1.pdf
Files
1503.03810v1.pdf
Additional details
Identifiers
- Eprint ID
- 71866
- Resolver ID
- CaltechAUTHORS:20161109-083152306
Related works
- Describes
- https://arxiv.org/abs/1503.03810 (URL)
Funding
- American Institute of Mathematics
- NSF
- DMS-1349399
- York University
- NSF
- DMS-1201435
Dates
- Created
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2016-11-09Created from EPrint's datestamp field
- Updated
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2021-11-11Created from EPrint's last_modified field