Minkowski Question Mark Function Research Articles

Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function

This paper continues investigations on the integral transforms of the Minkowski question mark function. In this work we finally establish the long-sought formula for the moments, which does not explicitly involve regular continued fractions, though it has a hidden nice interpretation in terms of semi-regular continued fractions. The proof is self-contained and does not rely on previous results by the author.

Addenda and corrigenda to “The Minkowski question mark function: explicit series for the dyadic period function and moments”

In this supplement we fully complete the proofs of the statements which were left out or only briefly sketched in the main paper.

THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

The Minkowski question mark function: explicit series for the dyadic period function and moments

Previously, several natural integral transforms of the Minkowski question mark function F(x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F(x). One of them - the dyadic period function G(z) - was defined as a Stieltjes transform. In this paper we introduce a family of distributions F_p(x) for Re p>=1, such that F_1(x) is the question mark function and F_2(x) is a discrete distribution with support on x=1. We prove that the generating function of moments of F_p(x) satisfies the three term functional equation. This has an independent interest, though our main concern is the information it provides about F(x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.

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 Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals.

An asymptotic formula for the moments of the Minkowski question mark function in the interval [0, 1

In this paper, we prove an asymptotic formula for the moments of the Minkowski question mark function, which describes the distribution of rationals in the Farey tree. The main idea is to demonstrate that a certain variation of a Laplace method is applicable in this problem, and hence the task reduces to a number of technical calculations.

Multidimensional continued fractions and a Minkowski function

The Minkowski question mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map.

Fractal analysis for sets of non-differentiability of Minkowski's question mark function

In this paper we study various fractal geometric aspects of the Minkowski question mark function Q. We show that the unit interval can be written as the union of the three sets Λ 0 : = { x : Q ′ ( x ) = 0 } , Λ ∞ : = { x : Q ′ ( x ) = ∞ } , and Λ ∼ : = { x : Q ′ ( x ) does not exist and Q ′ ( x ) ≠ ∞ } . The main result is that the Hausdorff dimensions of these sets are related in the following way: dim H ( ν F ) < dim H ( Λ ∼ ) = dim H ( Λ ∞ ) = dim H ( L ( h top ) ) < dim H ( Λ 0 ) = 1 . Here, L ( h top ) refers to the level set of the Stern–Brocot multifractal decomposition at the topological entropy h top = log 2 of the Farey map F, and dim H ( ν F ) denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with F. The proofs rely partially on the multifractal formalism for Stern–Brocot intervals and give non-trivial applications of this formalism.