On the Density of Hausdorff Dimensions of Bounded Type Continued Fraction Sets: The Texan Conjecture

Abstract

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

On the derivative of the Minkowski question mark function ?(x)

“…We note that Theorem 3 fails to hold if we assume that all partial quotients are bounded by 5. Here the numerical value of Hausdorff dimension for F 4 is taken from [1]. Some recent results on multifractal analysis of the sets associated with values of ?…”

A strengthening of a theorem of Bourgain and Kontorovich

“…Он, в частности, доказал [7] , что для простого d существует натуральное число b < d, такое что b d ∈ R A для алфавита A вида (1.2) при A log d. Пусть δ A хаусдорфова размерность множества бесконечных цепных дробей с непол-ными частными из произвольного конечного алфавита A. Бургейн и Конторович в 2011 году доказали, в частности, следующие две теоремы. Результаты Хенсли [2] дают веские основания предполагать, что неравенству (1.3) удовлетворяет алфавит (1.2) при A = 50 (но не A = 34 ввиду результатов Дженкин-сона [3] ).…”

“…Согласно результатам работы Дженкинсона [3] , неравенству (2.1) удовлетворяют все алфавиты вида A = {1, 2, 3, 4, n}, где число n может принимать любое из значений 6,7,8,9,10. 3 Благодарности.…”

Hausdorff dimensions of bounded-type continued fraction sets of Laurent series

“…A number of authors have considered the problem of calculating the Hausdorff dimension dim E A of E A ; see [3,4,[7][8][9][10][11]14,15,20]. Bounded-type continued fraction sets represent sets of numbers with certain diophantine approximation conditions and the Hausdorff dimensions have theoretical significance in terms of the Markoff and Lagrange spectra (see [5] [10]). Mauldin, Urbański [15] and Jenkinson [10] proved that D is dense in [0, 1 2 ]; also O. Jenkinson did some computations which did lend considerable support to the Texan conjecture.…”