论文标题
Cubic Laurent系列的持续分数
Continued fractions of cubic Laurent series
论文作者
论文摘要
我们在$ \ mathbb {q} [[t^{ - 1}] $中为Laurent系列的几个家庭构建了持续的分数扩展。据作者所知，这是此类结果的第一个结果，因为高斯得出了$（1+t）^r $，$ r \，$ r \ in \ Mathbb {q} $在1813年的持续扩展。作为一种应用，我们将超地域的类似物应用于其中一个家庭，并在其中一个家庭中，并在decort $ | $ | $ 3X^3-3tx^2-3ax+at $，$ a，t \ in \ mathbb {z} $和任何有理数的真正根源在参数$ a $ a $和$ t $的情况下，$ a，$ a，t \ in \ mathbb {z} $。我们还表明，\ Mathbb {r} $ in \ Mathbb {r}中的每个立方非理性$ X \允许以明确计算的封闭形式（广义）持续分数扩展。最后，我们提供了一系列无数的非理性$ x $，这些非理性$ x $是任意（但有限）许多比预期的理性近似值。也就是说，它们是这样的，以至于对于任何$τ<3+ \ frac {15 \ ln 2} {24} \ oft 3.4332 ... $不等式$ || qx || <（h（x）^τqe^{c \ sqrt {\ ln q}}}）^{ - 1} $在整数$ q $中具有许多解决方案。
We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction expansion for $(1+t)^r$, $r\in\mathbb{Q}$ in 1813. As an application, we apply an analogue of the hypergeometric method to one of those families and derive non-trivial lower bounds on the distance $|x - \frac{p}{q}|$ between one of the real roots of $3x^3 - 3tx^2-3ax+at$, $a,t\in\mathbb{Z}$ and any rational number, under relatively mild conditions on the parameters $a$ and $t$. We also show that every cubic irrational $x\in\mathbb{R}$ admits a (generalised) continued fraction expansion in a closed form that can be explicitly computed. Finally, we provide an infinite series of cubic irrationals $x$ that have arbitrarily (but finitely) many better-than-expected rational approximations. That is, they are such that for any $τ< 3+\frac{15\ln 2}{24}\approx 3.4332...$ the inequality $||qx|| < (H(x)^τ qe^{c\sqrt{\ln q}})^{-1}$ has many solutions in integer $q$.
