Math 574
نویسندگان
چکیده
1. Reference and definition The formulation of the “divided cell” algorithm as an approach to the study of the values of a function f (x, y) that is a product of two inhomogeneous linear forms on a lattice in R2 appears in E. S. Barnes, “The inhomogeneous minima of binary quadratic forms (IV)”, Acta Math. 92 (1954) 235–264. Although this is the fourth paper in a series, the first three were written with H. P. F. Swinnerton-Dyer, and the use of divided cells only appeared in the third paper. Thus, this paper can be considered to be the beginning of the systematic study. Subsequent work appeared in the dissertations of Peter Blanksby and Jane Pitman, written under the direction of Barnes. One of the results of Pitman is a striking proof of the existence of divided cells and a relation of this construction to the ordinary continued fraction algorithm. The significance of this construction emerged in conversations with David Crisp, William Moran, and Charles Pearce at the University of Adelaide. Our definitions differ slightly from those of Barnes since we want to allow degenerate cases.
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Topics in Logic and Foundations: Spring 2004
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