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The Limit Points of Weyl Sums and other Continuous Cocycles
Author(s) -
Forrest A. H.
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/54.3.440
Subject(s) - mathematics , limit (mathematics) , transcendental number , transitive relation , abelian group , pure mathematics , exponential function , series (stratigraphy) , dynamical systems theory , degree (music) , combinatorics , discrete mathematics , mathematical analysis , physics , quantum mechanics , paleontology , acoustics , biology
This paper examines the limit points of continuous Z‐cocycles defined on a minimal dynamical system and taking values in an abelian metrisable group. This is motivated by and applied to the study of classical exponential series Σ n = 1 Ne 2 π i ( n 2 θ + n x )to show, in particular, that if θ is a transcendental number with lim inf q q 3/2 ‖ q θ‖ < ∞, then for each x chosen from a dense G δ subset of [0, 1], these partial sums are dense in C. Indeed the associated cocycle is topologically transitive. Several generalisations to higher degree polynomials and higher dimensions are presented.