Mathematics Department
Permanent URI for this community
Browse
Browsing Mathematics Department by Author "Chaika, Jon"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Borel-Cantelli Sequences(Springer, 2012-06) Boshernitzan, Michael; Chaika, JonItem Dichotomy for arithmetic progressions in subsets of reals(American Mathematical Society, 2016) Boshernitzan, Michael; Chaika, JonLet 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).Item Zero measure spectrum for multi-frequency Schrödinger operators(EMS Press, 2022) Chaika, Jon; Damanik, David; Fillman, Jake; Gohlke, PhilippBuilding on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.