论文标题
马尔可夫和拉格朗日光谱的分形维度接近$ 3 $
Fractal dimensions of the Markov and Lagrange spectra near $3$
论文作者
论文摘要
Lagrange Spectrum $ \ Mathcal {l} $和Markov Spectrum $ \ Mathcal {M} $是真实线的子集，具有复杂的分形特性，在研究二磷酸的研究中自然而然地出现。众所周知，这些集合与任何半行重合的相交的Hausdorff尺寸，即$ \ Mathrm {dim} _ {\ Mathrm {h}}（\ Mathcal {l} \ Mathrm {dim} _ {\ Mathrm {h}}（\ Mathcal {M} \ cap（ - \ infty，t））：= D（t）$ = D（t）$，每$ t \ geq 0 $。还知道每一个$ \ varepsilon> 0 $ $ d（3）= 0 $和$ d（3+ \ varepsilon）> 0 $。 我们表明，对于$ \ varepsilon> 0 $的足够小的值，一个人具有近似值$ d（3+ \ varepsilon）= 2 \ cdot \ frac {w（e^{c_0} | \ log \ log \ log \ varepsilon |） \ varepsilon |}+\ mathrm {o} \ left（\ frac {\ log | \ log | \ log \ log \ varepsilon |} {| \ log \ log \ log \ varepsilon |^2} \ right）$，其中$ w $ w $ w $表示lambert函数$ c_0 = - \ log \ log（（3+ \ sqrt {5}）/2）\大约0.0383 $。我们还表明，通过“合理”函数，该结果对于$ d（3+ \ varepsilon）$的近似值是最佳的，从某种意义上说，如果$ f（t）$是$ c^2 $函数，以至于$ d（3+ \ varepsilon）= f（\ varepsilon）= f（\ varepsilon）+\ \ \ \ \ \ \ nog | \ varepsilon |} {| \ log \ varepsilon |^2} \ right）$，然后其第二个导数$ f''（t）$更改符号无限多次，因为$ t $ thut $ t $ toble $ 0 $。
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 $0$.
