Dichotomy for arithmetic progressions in subsets of reals | Zendy

Michael Boshernitzan | Zendy; Jon Chaika | Zendy

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Dichotomy for arithmetic progressions in subsets of reals

Author(s) -

Michael Boshernitzan,

Jon Chaika

Publication year - 2016

Publication title -

proceedings of the american mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.968

H-Index - 84

eISSN - 1088-6826

pISSN - 0002-9939

DOI - 10.1090/proc/13273

Subject(s) - mathematics , countable set , homeomorphism (graph theory) , image (mathematics) , nowhere dense set , discrete mathematics , borel set , set (abstract data type) , combinatorics , arithmetic , artificial intelligence , computer science , programming language

Let H stand for the set of homeomorphisms φ:[0, 1] → [0, 1]. We prove the following dichotomy for Borel subsets A ⊂ [0, 1]: • either there exists a homeomorphism φ ∈ Hsuch that the image φ(A) contains no 3-term arithmetic progressions; • or, for every φ ∈ H, the image φ(A) contains arithmetic progressions of arbitrary finite length. In fact, we show that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets)

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