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Keywords:
Lüroth expansion; optimal approximation; Hausdorff dimension
Lüroth expansion; optimal approximation; Hausdorff dimension
Summary:
For any $x\in (0,1]$, let $$ x=\frac {1}{d_1}+\frac {1}{d_1(d_1-1)d_2}+\dots +\frac {1}{d_1(d_1-1) \dots d_{n-1}(d_{n-1}-1)d_{n}}+\dots $$ be its Lüroth expansion. Denote by ${P_n(x)}/{Q_n(x)}$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents $\{{P_n(x)}/{Q_n(x)}\}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\{x\in (0,1]\colon |x-{P_n(x)}/{Q_n(x)}|<{1}/{Q_n(x)^{\nu +1}} \text{infinitely often}\}$ for any $\nu \ge 1$.
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