On the Frequency of Partial Quotients of Regular Continued Fractions
نویسندگان
چکیده
We consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.
منابع مشابه
Complex Numbers with Bounded Partial Quotients
Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hu...
متن کاملPalindromic continued fractions
An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion of algebraic real numbers of degree at least three. In particular, it is asked whether such numbers have or not arbitrarily large partial quotients in their continued fraction expansion. Although almost nothing has been proved yet in this direction, some more general speculations are due to La...
متن کاملContinued Fractions with Partial Quotients Bounded in Average
We ask, for which n does there exists a k, 1 ≤ k < n and (k, n) = 1, so that k/n has a continued fraction whose partial quotients are bounded in average by a constant B? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B = 2.
متن کاملContinued fractions of Laurent series with partial quotients from a given set
1. Introduction. Van der Poorten and Shallit's paper [10] begins: " It is notorious that it is damnably difficult to explicitly compute the continued fraction of a quantity presented in some other form ". The quantity is usually presented either as a power series or as the root of a specific equation. There has been some success in the former case for continued fractions of real numbers, such a...
متن کاملOn the Littlewood conjecture in simultaneous Diophantine approximation
For any given real number α with bounded partial quotients, we construct explicitly continuum many real numbers β with bounded partial quotients for which the pair (α, β) satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.
متن کامل