Abstract

Call a reduced fraction with denominator q a bronze, silver, or gold approximation with respect to a given irrational number ω if they differ by less than the reciprocal of the product of the square of q and 1, 2, or $\sqrt{5},$ respectively. Suppose A and B are neighboring Farey fractions sandwiching ω, where A’s denominator is at least as large as B’s denominator. Then B is bronze; either A or B is silver; and at least one but not all of A, B, and their mediant is gold. We prove afresh these results using the geometry of Ford circles rather than a purely algebraic approach, and explore some open questions with respect to the bronze-silver-gold classification while panning for gold fractions among the continued fraction convergents for ω.

摘要

如果分母q相差小于q与1、2或2的平方的乘积的倒数，则称分母为q的缩减分数是相对于给定无理数ω的青铜，银或金的近似值。$\sqrt{5},$分别。假设A和B是相邻的 Farey 分数，夹着ω，其中A的分母至少与B的分母一样大。那么B是青铜；或者甲或乙是银; 并且A，B至少一个但不是全部，它们的中位数是金。我们使用福特圆的几何学而不是纯粹的代数方法重新证明了这些结果，并探索了一些关于青铜-银-金分类的开放性问题，同时在ω 的连分数收敛中淘选金分数。