拉格朗日谱和马尔可夫谱上的新空白
New gaps on the Lagrange and Markov spectra
摘要：
令$L$和$M$分别表示拉格朗日谱和马尔可夫谱。已知$L\subset M$和$M\setminus L\neq\varnothing$。在这项工作中，我们展示了新的差距$L$和$M$使用两种方法。首先，我们通过描述接近3.938的$M\setminus L$的一个新部分来推导出这样的间隙：这个区域（以及其他三个候选区域）是通过研究最近由V. Delecroix和最后两位作者制作的$L$的图片，并借助本文附录之一中解释的算法发现的。作为副产品，我们还得到了$M\setminus L$的最大已知元素，我们改进了前两位作者与M. Pollicott和P.Vytnova（在$M\setminus L$的维数上，我们得到了$0.593$的一个新的下界）。其次，我们使用一个renormalisation的想法和厚度标准（让人想起第三作者的博士论文），以检测无限多的$M$积累到弗赖曼的差距之前，所谓的霍尔射线$[4.52782956616...,\infty)\subset L$的最大差距。
英文摘要：
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 $[4.52782956616...,\infty)\subset L$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
