Branch Mathematics and Statistics Faculty and Staff Publications
Document Type
Article
Publication Date
2-1-1998
Abstract
By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small w, to Cornu spirals (C-spirals), we prove the precise renormalization formula. This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map whose orbits are analyzed by expressing w as an even continued fraction.
Publisher
American Mathematical Society
Publication Title
Transactions of the American Mathematical Society
ISSN
0002-9947
Volume
350
Issue
2
First Page
615
Last Page
641
Language (ISO)
English
Recommended Citation
Transactions of the American Mathematical Society, 350(2): 615-641
Comments
Article author is part of the Main Campus Math Department.