Question Mark Function Research Articles

On the Derivative of the Minkowski Question-Mark Function

Abstract The Minkowski question-mark function ?(x) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ? (x)atthe point x =[0; a 1,a 2,...,a t ,...] is connected with the limit behaviour of the arithmetic mean (a 1 +a 2 +···+a t )/t. Particularly, N. Moshchevitin and A. Dushistova showed that if a 1 + a 2 + ⋯ + a t < κ 1 , {a_1} + {a_2} + \cdots + {a_t} < {\kappa _1}, where κ 1 = 2 log ( 1 + 5 2 ) / log 2 = 1.3884 … {\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots , then ?′(x)=+∞.They also proved that the constant κ 1 is non-improvable. We consider a dual problem: how small can be the quantity a 1 + a 2 + ··· + a t − κ 1 t if we know that ? (x) = 0? We obtain the non-improvable estimates of this quantity.

Attractors of dual continued fractions

Given a Farey-type map F with full branches in the extended Hecke group Γm±, its dual F♯ results from constructing the natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that the domain of F♯ is always tame, that is, it always contains intervals. As a main technical tool we construct, for every m=3,4,5,…, a homeomorphism Mm that simultaneously linearizes all maps with branches in Γm±, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicitly compute the Hölder exponent of every Mm, generalizing Salem's results for the Minkowski question mark function M3.

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Abstract Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn , x∈ X and a function f over X we consider the sums Sn (f, x) = ∑ g∈Gnf(gx). The asymptotic behaviour of Sn (f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n .

The derivative of the Minkowski function

We prove new results on the derivative of the Minkowski question mark function.

Calkin-Wilf Tree

In this article, we prove various properties of the Calkin-Wilf tree. We also see how the Minkowski question mark function acts on the Calkin-Wilf tree and its diagonals.

Billiards on pythagorean triples and their Minkowski functions

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincaré disk model. Our tables have $ m\ge3 $ ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group $ {\rm{PSU}}^\pm_{1,1} \mathbb{Z}[i] $. The resulting billiard map $ \widetilde B $ acts on the de Sitter space $ x_1^2+x_2^2-x_3^2 = 1 $, and has a natural factor $ B $ on the unit circle, the pythagorean triples appearing as the $ B $-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) $ B $-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over $ \mathbb{Q}(i) $.Each $ B $ as above is a $ (m-1) $-to-$ 1 $ orientation-reversing covering map of the circle, a property shared by the group character $ T(z) = z^{-(m-1)} $. We prove that there exists a homeomorphism $ \Phi $, unique up to postcomposition with elements in a dihedral group, that conjugates $ B $ with $ T $; in particular $ \Phi $ -whose prototype is the classical Minkowski question mark function- establishes a bijection between the set of points of degree $ \le2 $ over $ \mathbb{Q}(i) $ and the torsion subgroup of the circle. We provide an explicit formula for $ \Phi $, and prove that $ \Phi $ is singular and Hölder continuous with exponent $ \log(m-1) $ divided by the maximal periodic mean free path in the associated billiard table.

Diophantine properties of fixed points of Minkowski question mark function

We consider irrational fixed points of the Minkowski question mark function $? (x)$, that is irrational solutions of the equation $? (x)=x$. It is easy to see that there exist at least two such points. Although it is not known if there are other fixed points, we prove that the smallest and the greatest fixed points have irrationality measure exponent equal to 2. We give more precise results about the approximation properties of these fixed points. Moreover, in Appendix we introduce a condition from which it follows that there are only two irrational fixed points.

On the affirmative solution to Salem’s problem

Abstract The Salem problem to verify whether Fourier–Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier–Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier–Stieltjes coefficients of a power {m\in\mathbb{N}} of the Minkowski question mark function are derived.

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

The Minkowski question mark function [Formula: see text] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus [Formula: see text] is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

Alternating Sums of Elements of Continued Fractions and the Minkowski Question Mark Function

The paper considers the function A(t) (0 ≤ t ≤ 1), related to the distribution of alternating sums of elements of continued fractions. The function A(t) possesses many properties similar to those of the Minkowski function ?(t). In particular, A(t) is continuous, satisfies similar functional equations, and A′(t) = 0 almost everywhere with respect to the Lebesgue measure. However, unlike ?(t), the function A(t) is not monotonically increasing. Moreover, on any subinterval of [1, 0], it has a sharp extremum.

On Minkowski type question mark functions associated with even or odd continued fractions

We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.

Random Walks in the Hyperbolic Plane and the Minkowski Question Mark Function

Consider $$G=SL_2(\mathbb {Z})/\{\pm I\}$$ acting on the complex upper half plane H by $$h_M(z)=\frac{az\,+\,b}{cz\,+\,d}$$ for $$M \in G$$ . Let $$D=\{z \in H: |z|\ge 1, |\mathfrak {R}(z)|\le 1/2\}$$ . We consider the set $${\mathcal {E}} \subset G$$ with the nine elements M, different from the identity, such that $$\mathrm{tr\,}(MM^T)\le 3$$ . We equip the tiling of H defined by $$\mathbb {D}=\{h_M(D){:}\, M \in G\}$$ with a graph structure where the neighbours are defined by $$h_M(D) \cap h_{M'}(D) \ne \emptyset $$ , equivalently $$M^{-1}M' \in {\mathcal {E}}$$ . The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point X of the real line with the same distribution of $$S_2 W^{S_1}$$ , where $$S_1,S_2,W$$ are independent with $$\Pr (S_i=\pm 1)=1/2$$ and where W is valued in (0, 1) with distribution $$\Pr (W<w)=\mathbf ? (w)$$ . Here $$\mathbf ? $$ is the Minkowski function. If $$K_1, K_2, \ldots $$ are i.i.d with distribution $$\Pr (K_i=n)= 1/2^n$$ for $$n=1,2,\ldots $$ , then $$W= \frac{1}{K_1+\frac{1}{K_2+\ldots }}$$ : this known result (Isola in Appl Math 5:1067–1090, 2014) is derived again here.

Minkowski’s question mark measure

Minkowski’s question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman–Saff–Stahl–Totik and moreover that it belongs to a Nevai class; we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Möbius maps.

The modular group and words in its two generators*

Consider the full modular group $\sf{PSL}_{2}(\mathbb{Z})$ with presentation $\langle U,S|U^3,S^2\rangle$. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper might be considered as a necessary appendix), we are lead to the following natural question. Some words in the alphabet $\{U,S\}$ are equal to the unity; for example, $USU^3SU^2$ is such a word of length $8$, and $USU^3SUSU^3S^3U$ is such a word of length $15$. Given $n\in\mathbb{N}_{0}$. Find the number of words of length $n$ which are equal to the unity. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over $\mathbb{Q}(x)$ of degree $3$. As an aside, we formulate the problem of describing all algebraic functions with a Fermat property.

Derivatives of slippery Devil's staircases

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 increasing. The best known example of such a function is the Minkowski question-mark function, which was proved to be singular by Salem, in a paper which introduced some other constructions of singular functions. We describe all of these examples. Also we consider various generalisations of the Minkowski question-mark function, such as $α$-Farey-Minkowski functions. These examples all arise from one-dimensional dynamics. A few open questions and suggestions for filling minor gaps in the literature are proposed. Finally, we go back to ordinary Devil's staircases (i.e. non-decreasing singular functions) and discuss work done in that setting with the more general Hölder derivatives, and consider the outlook to extend those results to the strictly increasing situation.

DIFFERENTIABILITY AND NON-DIFFERENTIABILITY POINTS OF THE MINKOWSKI QUESTION MARK FUNCTION

Using the periodic continued fraction, we give concrete examples of the points at which the derivatives of the Minkowski question mark function does not exist. We also give examples of the differentiability points which show that recent apparently independent results are consistent and closely related.

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham's functional equations. Under some smoothness conditions of functions consisting the equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our results are applicable to some cases that the functions consisting the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply singularity of some well-known singular functions, in particular, Minkowski's question-mark function, and, some small perturbed functions of the singular functions.

Generalised Lüroth expansions and a family of Minkowski's question-mark functions

Minkowski's question-mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the twentieth century. Using different representations of real numbers by infinite sequences of integers, called α-Lüroth expansions, we obtain different instances of the standard shift map on infinite symbols, all of them topologically conjugated to the Gauss Map. In this note we prove that each of these conjugations share properties with Minkowski's question-mark function: all of them are singular homeomorphisms of the interval, and in the “rational” cases, they map the set of quadratic surds into the set of rational numbers. In this sense, this family is a natural generalisation of Minkowski's question-mark function.

Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure \(\mu \) for the dynamical system defined by the Gauss map \(x \mapsto 1/x \,\,{\mathrm {mod}}\,1\) is a Rajchman measure with polynomially decaying Fourier transform $$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$ We show that this property holds for any Gibbs measure \(\mu \) of Hausdorff dimension greater than 1 / 2 with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than 1 / 2 on badly approximable numbers, which extends the constructions of Kaufman and Queffelec–Ramare. Our main result implies that the Fourier–Stieltjes coefficients of the Minkowski’s question mark function decay to 0 polynomially answering a question of Salem from 1943. As an application of the Davenport–Erdős–LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman–Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.

The affirmative solution to Salem’s problem

The Salem problem to verify whether Fourier-Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier-Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier-Stieltjes coefficients of a power $m\in \mathbb{N}$ of the Minkowski question mark function are derived.