Raney变换器与$p$-拉格朗日谱的最低点
Raney Transducers and the Lowest Point of the $p$-Lagrange spectrum
AI总结：
针对刻画与自身p倍均极无理的实数问题，提出用Raney变换器计算p-拉格朗日谱最小值的算法，验证了p<2000时该值为有理数平方根，发现最高值对应Heegner素数且相关连分数满足对称关系。
摘要：
众所周知，黄金分割比$\phi$是“最无理的”数，因为其最佳有理逼近$s/t$的误差约为$1/(\sqrt{5} t^2)$，且常数$\sqrt{5}$是所能达到的最小值。给定素数$p$，如何刻画满足$x$和$px$均“非常无理”的实数$x$？这等价于寻找此前由第三作者定义的$p$-拉格朗日谱$\mathcal{L}_p$的最低点。我们描述了一种使用Raney变换器（Raney transducers）的算法，若该算法终止，即可计算$\min \mathcal{L}_p$，我们猜想它总能终止。我们验证了对于小于2000的素数$p$，$\min \mathcal{L}_p$是一个有理数的平方根。不可思议的是，$\min \mathcal{L}_p$的最高值出现在Heegner素数67、3和163上；且对所有$p$，对应的极无理数$x$和$px$的连分数均满足三种对称关系之一。
英文摘要：
It is well known that the golden ratio $ϕ$ is the ''most irrational'' number in the sense that its best rational approximations $s/t$ have error $\sim 1/(\sqrt{5} t^2)$ and this constant $\sqrt{5}$ is as low as possible. Given a prime $p$, how can we characterize the reals $x$ such that $x$ and $p x$ are both ''very irrational''? This is tantamount to finding the lowest point of the $p$-Lagrange spectrum $\mathcal{L}_p$ as previously defined by the third author. We describe an algorithm using Raney transducers that computes $\min \mathcal{L}_p$ if it terminates, which we conjecture it always does. We verify that $\min \mathcal{L}_p$ is the square root of a rational number for primes $p < 2000$. Mysteriously, the highest values of $\min \mathcal{L}_p$ occur for the Heegner primes $67$, $3$, and $163$, and for all $p$, the continued fractions of the corresponding very irrational numbers $x$ and $p x$ are in one of three symmetric relations.
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