Good Points for Diophantine Approximation
Daniel Berend Artūras Dubickas
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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 $|y-x_n| < \delta_n$ 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’.
Daniel Berend^{1} ^{} Artūras Dubickas^{2} ^{}
Volume 131, 2021
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