Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Abstract

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov 1 spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [31]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain-Kontorovich [2], Huang [18] and Kan [27]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [36].In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement. * The first author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1 the second author is partly supported by EPSRC grant EP/T001674/1.1 Markov's name will appear in this article in two contexts, namely those of Markov spectra in number theory and the Markov condition from probability theory. Since the both notions are associated with the same person (A.A. Markov, 1856(A.A. Markov, -1922, we have chosen to use the same spelling, despite the conventions often used in these different areas.

Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra

“…We subdivide each of the intervals J(R 1 ), J(R 2 ), and J(R 3 ) following the same process as before, with a separate decision tree in each case. In Subsection 4.6.3 the algorithm described in [15] will be implemented to rigorously establish the bound dim H X < 0.5 − 10 −8 . Summing up, we get the desired lower bound (2.7)…”

“…Ω claimed in the previous sections. In this section we would like to explain how to adapt the method developed in [15] to the present setting, to make the computation practical.…”

“…It is known that M \ L has zero Lebesgue measure. Furthermore, it was proved in [11] and [15] that the Hausdorff dimension of M \ L satisfies 0.5312 < dim H (M \ L) < 0.8823.…”

“…Following the approach developed by the first two authors [11], we use fine-grained combinatorial analysis of continued fractions to construct a cover M \ L by arithmetic sums of Gauss-Cantor sets and the so-called "Cantor sets of the gaps". We then apply the new method for computing the Hausdorff dimension recently developed by the last two authors [15] to several Gauss-Cantor sets to obtain sharper upper bounds on dim H (M \ L).…”

“…Next §3 is dedicated to the construction and analysis of the arithmetic sums of Gauss-Cantor sets and "Cantor sets of the gaps" which cover M \ L, and subsequently allows us to obtain an upper bound on dim H (M \ L). The intricate character of the Gauss-Cantor sets involved in estimates in §2 and §3 means that the algorithm for computing the Hausdorff dimension developed in [15] has to be considerably adapted and improved. For completeness, in §4 we explain how the Hausdorff dimension of the complicated Gauss-Cantor sets can be computed and give some details of the numerical implementation.…”