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

@article{Dushistova2007OnTD,
title={On the derivative of the Minkowski question mark function ?(x)},
author={Anna Aleksandrovna Dushistova and Nikolai Germanovich Moshchevitin},
journal={Journal of Mathematical Sciences},
year={2007},
volume={182},
pages={463-471}
}
• Published 15 June 2007
• Mathematics
• Journal of Mathematical Sciences
Let x = [0; a1, a2, …] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ?′(x) = +∞, provided that $\mathop {{\lim \sup }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} < {\kappa_1} = \frac{{2\log {\lambda_1}}}{{\log 2}} = {1.388^{+} }$, and ?′(x) = 0, provided that $\mathop {{\lim \inf }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} > {\kappa_2} = \frac{{4{L_5} - 5{L_4… Number theoretic applications of a class of Cantor series fractal functions. I AbstractSuppose that $${{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}$$(P,Q)∈N2N×N2N and x = E0.E1E2 · · · is the P-Cantor series expansion of$${x \in On the derivative of iterations of the Minkowski question mark function at special points • N. Shulga • Mathematics Functiones et Approximatio Commentarii Mathematici • 2021 For the Minkowski question mark function ?(x) we consider derivative of the function fn(x) = ?(?(...? } {{ } n times (x))). Apart from obvious cases (rational numbers for example) it is non-trivial V Tree -- Continued Fraction Expansion, Stern-Brocot Tree, Minkowski's$?(x)$Function In Binary: Exponentially Faster The Stern-Brocot tree and Minkowki's question mark function$?(x)$(or Conway's box function) are related to the continued fraction expansion of numbers from Q with unary encoding of the partial On the Minkowski Measure The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative The Minkowski question mark function: explicit series for the dyadic period function and moments It is proved that the dyadic period function G(z) is a sum of infinite series of rational functions with rational coefficients, including the question mark function F(x) and the generating function of moments of F_p(x). Minkowski question mark function and its generalizations, associated with p-continued fractions: fractals, explicit series for the dyadic period function and moments Previously, several natural integral transforms of Minkowski question mark function F (x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Derivatives of slippery Devil's staircases • Mathematics • 2017 In this paper we first give a survey of known results on the derivative of slippery Devil's staircase functions, that is, functions that are singular with respect to the Lebesgue measure and strictly Existence theory for semilinear evolution inclusions involving measures • Mathematics • 2017 We provide existence results for semilinear differential inclusions involving measures: 0.1 du∈Audt+F(t,u)dg,t∈[0,1],u(0)=u0,where A is the infinitesimal generator of a C0‐semigroup {T(t),t≥0} of ## References SHOWING 1-10 OF 12 REFERENCES The Derivative of Minkowski's ?(x) Function • Mathematics • 2001 Abstract Minkowski's ?( x ) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it enables us to ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE 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 A new light on Minkowski's?$(x)$function • Mathematics • 1998 The well--known Minkowski's?$(x)\$ function is presented as the asymptotic distribution function of an enumeration of the rationals in (0,1] based on their continued fraction representation. Besides,
On some singular monotonic functions which are strictly increasing
almost everywhere, may be constant in every interval contiguous to a perfect set of measure zero: it is usually said, in this case, that f(x) is of the Cantor type. There are, however, monotonic
A refinement of the rule of comparing continuants
We prove a theorem, which allow us to compare in some cases the values of continuants of denominators of finite continued fractions without their calculation. For natural numbers a { , α2 , . . . ,
Note on a singular fun tion of Minkowski .
• / / Pro . Amer . Math . So .
• 1960
On some singular monotone fun tions whi h are stri tly in reasing
• / / Trans . Amer . Math . So .
• 1943