论文标题
负分数为负分数的双苯胺近似
Diophantine approximation by negative continued fraction
论文作者
论文摘要
我们表明，分母的增长率$ n $ th $ th $ th $ th $ th $ x $的收敛性和近似速率：$$ \ frac {\ log {n}} {n} {n} \ log log {\ log {\ left | x- x- \ frac {p_n} {q_n} {q_n} {q_n} {q_n} {q_n} {q__n} {q_ right | right | right | - \ frac {π^2} {3} \ quad \ text {in Meath。} $$ for A.E. $ x $。在证明过程中，我们谴责已知的启发性结果，即负分数的数字算术平均值在量度上收敛到3，尽管极限下限为2，而上限上的极限几乎是无限的。
We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{π^2}{3} \quad \text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
