论文标题
拉格朗日和马尔可夫光谱上的新差距
New gaps on the Lagrange and Markov spectra
论文作者
论文摘要
令$ L $和$ M $分别表示Lagrange和Markov Spectra。众所周知，$ l \ subset m $和$ m \ setminus l \ neq \ varnothing $。在这项工作中，我们使用两种方法展示了$ L $和$ M $的新空白。首先，我们通过描述$ m \ setminus l $的新部分接近3.938：通过调查V. delecroix最近制作的$ l $的图片和最后两位作者的图片，以及本算法对本文的附录中解释了该算法的帮助。作为副产品，我们还获得了$ m \ setminus l $的最大已知元素，并且我们在过去的两个作者获得的$ m \ setminus l $上的下限上有所改善。其次，我们使用重态化的想法和厚度标准（让人联想到第三作者的博士学位论文）来检测$ m $累积的最大差距，这些间隙累积到弗雷曼的间隙，在所谓的Hall的Ray $ [4.522782956566616 ...，\ infty）之前。
Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. In this work, we exhibit new gaps of $L$ and $M$ using two methods. First, we derive such gaps by describing a new portion of $M\setminus L$ near to 3.938: this region (together with three other candidates) was found by investigating the pictures of $L$ recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of $M\setminus L$ and we improve upon a lower bound on the Hausdorff dimension of $M\setminus L$ obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of $0.593$ on the dimension of $M\setminus L$). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of $M$ accumulating to Freiman's gap preceding the so-called Hall's ray $[4.52782956616...,\infty)\subset L$.
