Farey Sequence Research Articles

New Ways to Balance Numbers

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from 1 to b – 1 is equal to the sum of the numbers from b + 1 to b + r for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.

Purely periodic continued fractions and graph-directed iterated function systems

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

Tutte polynomial for a small world connected copies of Farey graphs

Tutte polynomial for a small world connected copies of Farey graphs

Cycles in Austrian Solitaire

Austrian Solitaire is a variation of Bulgarian Solitaire. It may be described as a card game, a method of asset inventory management, or a discrete dynamical system on integer partitions. We prove that the limit cycles in Austrian Solitaire do not depend on the initial configuration; in other words, each state space is connected. We show that a full Farey sequence completely characterizes these unique (and balanced) cycles.

Smallest denominators

AbstractWe establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics.

On the distribution of index of Farey sequences

On the distribution of index of Farey sequences

Characterization of a spring pendulum phase-space trajectories.

We study the geometrical properties of phase-space trajectories (or orbits) of a spring pendulum as functions of the energy. Poincaré maps are used to describe the properties of the system. The points in the Poincaré maps of regular orbits (non-chaotic) cluster around separated segments or in chains of islands. Looking at how segments are formed, we conclude that the orbits are closely related to torus knots. Examining the toroidal and poloidal turns of the orbits, we introduce the definition of a rational parameter Ω, which is closely related to the concept of frequency used in the analysis of dynamical systems. Algorithms were developed to calculate Ω, and we found that this parameter naturally describes the orbits in terms of Farey sequences; also, calculations show that orbits with the same Ω have similar dynamics. Orbits corresponding to chains of islands are identified with cable knots that can be characterized using two parameters analogous to Ω. In some cases, non-trivial cable knots were found. With the analysis presented in this study, it is shown that Ω follows predictable distributions in the (z,Ω) space.

Joint partial equidistribution of Farey rays in negatively curved manifolds and trees

Abstract We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

Rainbow Connection Number on Generalized Farey Graph

Rainbow Connection Number on Generalized Farey Graph

The immersion-minimal infinitely edge-connected graph

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor.By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.

RELATIVE INTONATION: NON-SYMMETRICAL IMPLICATIONS OF LINEAR AND LOGARITHMIC INTERVALLIC MEASUREMENT

AbstractThis article investigates intervallic measurement and the tacit limitations engendered by a prevalent symmetrical perspective of measuring intervals. Various numerical and instrumental limitations and further detail of harmonic and melodic structures, such as Farey sequences, are illustrated. This approach distinguishes itself from a perspective of prime limits, explored by Harry Partch and others. A standardisation of ‘microtonal’ notation is not suggested; rather, the restrictions provided by any such standardisation are re-examined through an objective lens of ratios, to harness the generative potential of numbers. An orchestration-led approach to composition is described, where the tuning limitations of instruments are utilised for idiomatic composition. Tuning practices that ‘evade’ the octave are also discussed, including gamelan, mbira and three scales found by Wendy Carlos. The article concludes with a section on the construction of harmonic systems in the absence of instrumental influences.

Shrinking target horospherical equidistribution via translated Farey sequences

For a certain diagonal flow on SL(d,Z)﹨SL(d,R) where d≥2, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for d=2 and L﹨PSL(2,R) where L is any cofinite Fuchsian group with at least one cusp.The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.

Flip paths between lattice triangulations

We present a O(n32)-time algorithm for the shortest (diagonal) flip path problem for lattice triangulations with n points, improving over previous O(n2)-time algorithms. For a large, natural class of inputs, our bound is tight in the sense that our algorithm runs in time linear in the number of flips in the output flip path. Our results rely on an independently interesting structural elucidation of shortest flip paths as the linear orderings of a unique partially ordered set, called a minimum flip plan, constructed by a novel use of Farey sequences from elementary number theory. Flip paths between general (not necessarily lattice) triangulations have been studied in the combinatorial setting for nearly a century. In the Euclidean geometric setting, finding a shortest flip path between two triangulations is NP-complete. However, for lattice triangulations, which are studied as spin systems, there are known On2-time algorithms to find shortest flip paths. These algorithms, as well as ours, apply to constrained flip paths that ensure a set of constraint edges are present in every triangulation along the path. Implications for determining simultaneously flippable edges, i.e. finding optimal simultaneous flip paths between lattice triangulations, and for counting lattice triangulations are discussed.

On a problem of Steinhaus

AbstractLet be a positive integer. A sequence of points in the unit interval [0,1) is piercing if holds for every and every . In 1958, Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most 74 elements. This was later improved to the best possible value of 17 by Warmus, and independently by Berlekamp and Graham. In this paper, we study a more general variant of piercing sequences. Let be an infinite nondecreasing sequence of positive integers. A sequence is ‐piercing if holds for every and every . A special case of , with a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each , the maximum length of any ‐piercing sequence is finite. Expressing this maximum length as , they obtained an exponential upper bound on the function , which was later improved to by Graham and Levy. Recently, Konyagin proved that holds for all sufficiently big . Using a different technique based on the Farey fractions and stick‐breaking games, we prove here that the function satisfies where and . We also prove that there exists an infinite ‐piercing sequence with if and only if .

ON QUADRATIC FIELDS GENERATED BY POLYNOMIALS

AbstractLet $f(X) \in {\mathbb Z}[X]$ be a polynomial of degree $d \ge 2$ without multiple roots and let ${\mathcal F}(N)$ be the set of Farey fractions of order N. We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields ${\mathbb Q}(\sqrt {f(r)})$ for $r\in {\mathcal F}(N)$ , with a given discriminant.

On Nordhaus-Gaddum type relations of δ-complement graphs

The δ-complement graphs were introduced by Amrithalakshmi et al. in 2022. In their work, some interesting properties of the graphs such as δ-self-complementary, adjacency, and hamiltonicity were studied. In this work, we study the coloring aspect of the δ-complement graphs. In particular, we provide lower and upper bounds on the product and the summation between the chromatic number and the δ-chromatic number of a graph, in the same fashion as the well-known Nordhaus-Gaddum type relations. Classes of graphs that achieve those bounds are also given. Furthermore, we provide upper bounds on δ-chromatic numbers in terms of the clique numbers and compute the δ-chromatic numbers of certain graphs including ladder graphs, path graphs, complete m-partite graphs, and small-world Farey graphs.

Some Characteristics of the Farey Sequences with Ford Circles

Farey sequence is a pattern of rational numbers that approximates irrational numbers. In this paper, we use the Farey sequence to describe the Ford Circles. Its results and applications are equally fascinating as its pattern. Hurwitz Theorem is the main outcome of the approximation of irrational by rational numbers. Also, we examine the relationship between the Ford circle and the Farey sequence for rational values between 0 and 1.

Farey graph and rational fixed points of the extended modular group

Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group ¯¯¯¯ΓΓ¯ is the group of 2×22×2 matrices with integer entries and determinant ±1±1. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set ^Q=Q∪{∞}Q^=Q∪{∞}. In this study, we consider the elements in ¯¯¯¯ΓΓ¯ that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in $\overline{\Gamma}$ that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.

On the Degree Distribution of Haros Graphs

Haros graphs are a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary tree. Moreover, an expression that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

Farey tree and devil’s staircase of frequency-locked breathers in ultrafast lasers

Nonlinear systems with two competing frequencies show locking or resonances. In lasers, the two interacting frequencies can be the cavity repetition rate and a frequency externally applied to the system. Conversely, the excitation of breather oscillations in lasers naturally triggers a second characteristic frequency in the system, therefore showing competition between the cavity repetition rate and the breathing frequency. Yet, the link between breathing solitons and frequency locking is missing. Here we demonstrate frequency locking at Farey fractions of a breather laser. The winding numbers exhibit the hierarchy of the Farey tree and the structure of a devil’s staircase. Numerical simulations of a discrete laser model confirm the experimental findings. The breather laser may therefore serve as a simple test bed to explore ubiquitous synchronization dynamics of nonlinear systems. The locked breathing frequencies feature a high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.