Ambiguous Numbers
Problem 198
A best approximation to a real number for the denominator bound is a rational number (in reduced form) with , so that any rational number which is closer to than has .
Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. has the two best approximations and for the denominator bound .We shall call a real number ambiguous, if there is at least one denominator bound for which possesses two best approximations. Clearly, an ambiguous number is necessarily rational.
How many ambiguous numbers , are there whose denominator does not exceed ?