Triangles in diophantine approximation

JOURNAL OF NUMBER THEORY

卷:201

Article

Mundici, Daniele¹ 

[1] Univ Florence, Dept Math & Comp Sci Ulisse Dini, Viale Morgagni 67, I-50134 Florence, Italy

关键词: Regular cone;    Unimodular cone;    Fan;    Continued fraction expansion;    Simultaneous diophantine approximation;    Multidimensional continued fraction algorithm;    Stellar operation;    Starring;    Farey mediant;    Farey sum;    Davenport-Mahler theorem;   

DOI  :  10.1016/j.jnt.2019.02.011

来源: Elsevier

【 摘 要 】

For any point x = (x(1), x(2)) is an element of R-2 we let G(x) = Zx(1) + Zx(2) + Z be the subgroup of the additive group R generated by x(1), x(2), 1. When rank(G(x)) = 3 we say that x is a rank 3 point. We prove the existence of an infinite set I subset of R-2 of rank 3 points having the following property: For every two-dimensional continued fraction expansion mu and x is an element of I, letting mu(x) = T-0 superset of T-1 superset of ... , it follows that infinitely many triangles T-n have some angle <= arcsin(23(1/2)/6) approximate to pi/(3.3921424) approximate to 53 degrees. Thus lim inf(n ->infinity) area(T-n)/diam(T-n)(2) <= 23(1/2)/12. At the opposite extreme, we construct a two-dimensional continued fraction expansion mu and a dense set D subset of R-2 of rank 3 points such that for each x is an element of D the sequence T-0 superset of T-1 superset of ... of triangle of mu(x) has the following property: Letting omega(n) denote the smallest angle of T-n, it follows that omega(0) < omega(1) < ... and lim(n ->infinity) omega(n) = pi/3. Further, the other two angles of T-n are > pi/3. Thus lim(n ->)infinity area(T-n)/diam(T-n)(2) = 3(1/2)/4, and the vertices of the triangles T-n strongly converge to x. Our proofs combine a classical theorem of Davenport and Mahler with binary stellar operations of regular fans. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free