3附近的马尔可夫谱与拉格朗日谱的分形维数
Fractal dimensions of the Markov and Lagrange spectra near $3$
AI总结：
该研究针对3附近拉格朗日谱与马尔可夫谱的半直线交集豪斯多夫维数$d(3+\varepsilon)$，给出含Lambert函数的渐近近似式，并证明该近似在$C^2$函数逼近意义下的最优性。
摘要：
拉格朗日谱$\u2112$与马尔可夫谱$\u2133$是实数轴上具有复杂分形性质的子集，在丢番图逼近研究中自然出现。已知这两个集合与任意半直线的交集的豪斯多夫维数一致，即对所有$t \geq 0$，有$\mathrm{dim}_{\mathrm{H}}(\u2112 \cap (-\infty, t)) = \mathrm{dim}_{\mathrm{H}}(\u2133 \cap (-\infty, t)):= d(t)$。同时已知$d(3)=0$，且对任意$\varepsilon>0$，$d(3+\varepsilon)>0$。\n 我们证明，当$\varepsilon > 0$足够小时，有近似式$d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$，其中$W$表示Lambert函数（即$f(x)=xe^x$的反函数），$c_0=-\log\log((3+\sqrt{5})/2) \approx 0.0383$。我们还证明该结果在用“合理”函数逼近$d(3+\varepsilon)$的意义下是最优的：若$F(t)$是$C^2$函数，满足$d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$，则当$t$趋近于0时，其二阶导数$F''(t)$会无限次变号。
英文摘要：
The Lagrange spectrum $\mathcal{L}$ and Markov spectrum $\mathcal{M}$ are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimension of the intersection of these sets with any half-line coincide, that is, $\mathrm{dim}_{\mathrm{H}}(\mathcal{L} \cap (-\infty, t)) = \mathrm{dim}_{\mathrm{H}}(\mathcal{M} \cap (-\infty, t)):= d(t)$ for every $t \geq 0$. It is also known that $d(3)=0$ and $d(3+\varepsilon)>0$ for every $\varepsilon>0$. We show that, for sufficiently small values of $\varepsilon > 0$, one has the approximation $d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$, where $W$ denotes the Lambert function (the inverse of $f(x)=xe^x$) and $c_0=-\log\log((3+\sqrt{5})/2) \approx 0.0383$. We also show that this result is optimal for the approximation of $d(3+\varepsilon)$ by "reasonable" functions, in the sense that, if $F(t)$ is a $C^2$ function such that $d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)$, then its second derivative $F''(t)$ changes sign infinitely many times as $t$ approaches $0$.
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