A monad measure space for logarithmic density

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...

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 181; no. 3; pp. 577 - 599
Main Authors Di Nasso, Mauro, Goldbring, Isaac, Jin, Renling, Leth, Steven, Lupini, Martino, Mahlburg, Karl
Format Journal Article
LanguageEnglish
Published Vienna Springer Vienna 01.11.2016
Springer Nature B.V
Subjects
Density
Mathematics
Mathematics and Statistics
Theorems
Nonstandard analysis
Secondary 26E35
Primary 11B05
Logarithmic density
37A45
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Summary: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.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-016-0966-1