Badly Approximable

Every irrational number  has an approximation constant  defined by

where  is the nearest integer to  and  is the infimum limit. The quantity  measures how well  is approximated by the rational number .

 is said to be badly approximable if . An irrational number is badly approximable iff the terms  of its continued fraction  are bounded. Since quadratic surds have periodic continued fractions, they are badly approximable. The maximum possible value of  is , attained for example at , where  is the golden ratio.

REFERENCES:

Burger, E. B. Exploring the Number Jungle: A Journey Into Diophantine Analysis. Providence, RI: Amer. Math. Soc., 2000.