Good points for diophantine approximation Academic Article uri icon

abstract

  • Given a sequence (x n ) n=1 ∞ of real numbers in the interval [0, 1) and a sequence (δ n ) n=1 ∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x n | < δ n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x n ) n=1 ∞ , which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ n ) n=1 ∞ converges to zero. On the other hand, for any sequence of positive numbers (δ n ) n=1 ∞ tending to zero, there is a well distributed sequence (x n ) n=1 ∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.

publication date

  • January 1, 2009