Geometric properties of the Markov and Lagrange spectra. (English) Zbl 1404.11095

Moreira, Carlos

Ann. Math. (2) 188, No. 1, 145-170 (2018).

Denote by \(L\) the Lagrange spectrum and by \(M\) the Markoff spectrum. Both are closed subsets of the half line \([\sqrt 5,\infty)\) and \(L\) is a subset of \(M\). The Hall ray \([c,\infty)\) is the biggest half line contained in \(L\) and it is the same for \(M\). While \(L\cap [\sqrt 5,3)=M\cap [\sqrt 5,3)\) consists of an increasing sequence with limit \(3\), the sets \(L\cap (3,c)\) and \(M\cap (3,c)\), which are distinct, are not so well known. The set \((M\setminus L)\cap [\sqrt 5,3)\) have recently been investigated in preprints from the a uthor and C. Matheus [“\(HD(M\setminus L)>0.353\)”, Preprint, arXiv:1703.04302; “Markov spectrum near Freiman’s isolated points in \(M\setminus L\)”, Preprint, arXiv:1802.02454; “New numbers in \(M\setminus L\) beyond \(\sqrt{12}\): solution to a conjecture of Cusick”, Preprint, arXiv:1803.01230]. In the paper under review, the author goes much further and proves a number of deep new results related with the Hausdorff dimensions of \(L\cap (-\infty,t)\) and \(M\cap (-\infty,t)\) for \(t\in{\mathbb{R}}\). In particular, he answers several questions raised by Y. Bugeaud [Unif. Distrib. Theory 3, No. 2, 9–20 (2008; Zbl 1212.11071)] on these sets. He proves that the Lagrange spectrum \(L\) is a perfect set: the derivative \(L'\) is equal to the second derivative \(L''\); the question whether \(M''=M'\) is yet unsolved. Several other open problems are stated. One main tool is from a joint work of the author with J.-C. Yoccoz [Ann. Math. (2) 154, No. 1, 45–96 (2001; Zbl 1195.37015)] on regular Cantor sets.

Reviewer: Michel Waldschmidt (Paris)

MSC:

11J06	Markov and Lagrange spectra and generalizations
11J70	Continued fractions and generalizations
28A78	Hausdorff and packing measures
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Keywords:

Markov and Lagrange spectra; continued fractions; fractal dimensions; regular Cantor sets

Citations:

Zbl 1212.11071; Zbl 1195.37015

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