Approximation of real numbers with rational number sequences

Abstract:

Let $\alpha \in \mathbb {R}$, and let $C>\max \{1,\alpha \}$. It is shown that if $\{p_n/q_n\}$ is a sequence formed out of all rational numbers $p/q$ such that \begin{equation*} \left |\alpha -\frac {p}{q} \right | \leq \frac {1}{Cq^2}, \end{equation*} where $p\in \mathbb {Z}$ and $q\in \mathbb {N}$ are relatively prime numbers, then either $\{p_n/q_n\}$ has finitely many elements or \begin{equation*} \limsup _{n\to \infty }\frac {\log \log q_n}{\log n}\geq 1, \end{equation*} where the points $\{q_n\}_{n\in \mathbb {N}}$ are ordered by increasing modulus. This implies that the sequence of denominators $\{q_n\}_{n\in \mathbb {N}}$ grows exponentially as a function of $n$, and so the density of rational numbers which approximate $\alpha$ well in the above sense is relatively low.