• Daniel Berend

Articles written in Proceedings – Mathematical Sciences

• Good Points for Diophantine Approximation

Given a sequence $(x_n)^\infty_{n=1}$ of real numbers in the interval [0,1) and a sequence $(\delta_n)^\infty_{n=1}$ 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 $y\in[0,1]$ for which the inequality $|y-x_n| &lt; \delta_n$ holds for infinitely many positive integers 𝑛. We show that the set of well approximable’ points by a sequence $(x_n)^\infty_{n=1}$, which is dense in [0,1], is quite large’ no matter how fast the sequence $(\delta_n)^\infty_{n=1}$ converges to zero. On the other hand, for any sequence of positive numbers $(\delta_n)^\infty_{n=1}$ tending to zero, there is a well distributed sequence $(x_n)^\infty_{n=1}$ in the interval [0,1] such that the set of well approximable’ points 𝑦 is `quite small’.

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019