A walk along the branches of the extended Farey Tree

Lagarias, Jeffrey C.; Tresser, Charles

doi:10.1147/rd.393.0283

Cited by 28 publications

(23 citation statements)

References 27 publications

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“…A by far not complete overview of the papers written about the Minkowski question mark function or closely related topics (Farey tree, enumeration of rationals, Stern's diatomic sequence, various 1-dimensional generalizations and generalizations to higher dimensions, statistics of denominators and Farey intervals, Hausdorff dimension and analytic properties) can be found in [1]. These works include [5], [6], [8], [9], [10], [12], [13] (this is the only paper where the moments of a certain singular distribution, a close relative of F (x), were considered), [11], [14], [16], [18], [20], [24], [25], [26], [27], [28], [29], [30], [31], [33]. The internet page [36] contains an up-to-date and exhaustive bibliographical list of papers related to the Minkowski question mark function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Alkauskas

2010

Math. Comp.

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Abstract. 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 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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The holomorphicity of G p (z) follows exactly as in the case p = 1 [1]. All we need is the first integral in (20) and the fact that I p is a closed set.…”

Section: Three-term Functional Equationmentioning

confidence: 93%

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

Alkauskas

2010

Math. Comp.

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“…A fraction m/n is simpler than p/q if n ≤ q and m ≤ p, with at least one inequality strict [4, p. 166]. The methods described could be used on an extended Farey tree [5] over all rational numbers.…”

Section: Fraction Interpolationmentioning

confidence: 99%

“…A Farey series may be constructed beginning with 0/1 and 1/1 and then recursively inserting the mediant of adjacent fractions when the mediant denominator is of order n or less. A Farey tree [4,5] is the organization of proper fractions with 0/1 and 1/1 in an ancestry tree. Each proper fraction has two parents, the simplest parent (dotted line in our figures) and the closest parent (solid line in our figures) [4] (Lagarias and Tresser [5] call them old and young parents, respectively).…”

Section: Fraction Interpolationmentioning

confidence: 99%

Fraction interpolation walking a Farey tree

Mosko

Garcia-Luna-Aceves

2006

Information Processing Letters

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We present an algorithm to find a proper fraction in simplest reduced terms between two reduced proper fractions. A proper fraction is a rational number m/n with m < n and n > 1. A fraction m/n is simpler than p/q if m ≤ p and n ≤ q, with at least one inequality strict. The algorithm operates by walking a Farey tree in maximum steps down each branch. Through monte carlo simulation, we find that the present algorithm finds a simpler interpolation of two fractions than using the Euclidean-Convergent [1] walk of a Farey tree and terminating at the first fraction satisfying the bound. Analysis shows that the new algorithms, with very high probability, will find an interpolation that is simpler than at least one of the bounds, and thus take less storage space than at least one of the bounds.

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“…For θ > 0 the set of allowable symbol sequences for the additive expansion is the full one-sided shift on two letters {R, D}. A relation of this expansion to paths in the Farey tree is described in Lagarias [29] and Lagarias and Tresser [32]. We start with the interval [ At each step, we multiply our current matrix p 1 q 1 p 2 q 2 on the left by R or D, as appropriate.…”

Section: Additive Continued Fraction Expansionsmentioning

confidence: 99%

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

Grabiner

Lagarias

2001

Mh Math

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This paper describes the cutting sequences of geodesic flow on the modular surface H/P SL(2, Z) with respect to the standard fundamental domain F = {z = x + iy : − 1 2 ≤ x ≤ 1 2 and |z| ≥ 1} of P SL(2, Z). The cutting sequence for a vertical geodesic {θ + it : t > 0} is related to a onedimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L,R,J} which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain F up to an isometry of the hyperbolic plane H. We give conversion methods between the cutting sequence for the vertical geodesic {θ + it : t > 0}, the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first ℓ symbols when given as input the first O(ℓ) symbols of the ACF expansion, which takes time O(ℓ 2 ) and space O(ℓ).

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A walk along the branches of the extended Farey Tree

Cited by 28 publications

References 27 publications

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

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

Fraction interpolation walking a Farey tree

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

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