Algebraic Irrationalities Research Articles

Second bounded cohomology of groups acting on 1-manifolds and applications to spectrum problems

We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of 1-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group G0 constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to a homological version of the von Neumann–Day Problem, posed by Calegari. Next, we develop a technical refinement of our criterion to demonstrate the existence of finitely generated non-indicable (even simple) left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Navas. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Finally, we provide the first examples of manifolds whose simplicial volumes are algebraic and irrational, as further evidence towards a conjecture of Heuer and Löh.

Chaotic renormalization group flow and entropy gradients over Haros graphs.

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator R over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterization of low-dimensional nonlinear dynamics and has a renormalization group (RG) structure. We find that the dynamics of R over Haros graphs is complex and includes unstable periodic orbits of arbitrary period and nonmixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is associated with the set of rational numbers, and find periodic RG orbits that relate to (pure) quadratic irrationals and aperiodic RG orbits, related with (nonmixing) families of nonquadratic algebraic irrationals and transcendental numbers. Finally, we show that the graph entropy of Haros graphs is globally decreasing as the RG flows towards its stable fixed point, albeit in a strictly nonmonotonic way, and that such graph entropy remains constant inside the periodic RG orbit associated to a subset of irrationals, the so-called metallic ratios. We discuss the possible physical interpretation of such chaotic RG flow and put results regarding entropy gradients along RG flow in the context of c-theorems.

About Minimal Polynomial of Residual Fractions for Algebraic Irrationalities

About Minimal Polynomial of Residual Fractions for Algebraic Irrationalities

On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field ofpp-adic numbersQp\mathbb {Q}_p. MCFs have also been recently introduced inQp\mathbb {Q}_p, including, in particular, app-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of thepp-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finitepp-adic MCF representsQ\mathbb Q-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF havepp-adic valuations equal to−1-1. Finally, we show that a periodic MCF of dimensionmmconverges to an algebraic irrational of degree less than or equal tom+1m+1; for the casem=2m=2, we are able to give some more detailed results.

Unimodular Invariance of Karyon Expansions of Algebraic Numbers in Multidimensional Continued Fractions

By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.

Approximations of Algebraic Irrationalities with Matrices

We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic irrationalities giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we briefly perform a numerical comparison with well-known iterative methods (such as Newton and Halley ones), showing that the approximations provided by regular representations appear more accurate for the same size of the denominator.

КЛАССИФИКАЦИЯ ЧИСТО-ВЕЩЕСТВЕННЫХ АЛГЕБРАИЧЕСКИХ ИРРАЦИОНАЛЬНОСТЕЙ

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points.

Linear recurrence sequences and periodicity of multidimensional continued fractions

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. We provide a characterization for periodicity of Jacobi–Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.

On some infinite sums of iterated radicals

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals.

Pythagorean approximations and continued fractions

In this article, we will show that the Pythagorean approximations of 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.

Periodicity and Predictability in Chaotic Systems

Most of the good mathematical models available for natural phenomena are non linear, and these are often chaotic. Sensitive dependence on initial conditions can foil any attempt at prediction. In fact, measurement errors in collecting the initial data, intrinsic to any observational method, will propagate under the computational use of the model. With this, the final results obtained from the model may be useless when compared with the real future of the phenomena. Even in the strictly computational context, the sensitive dependence on initial conditions often makes a chaotic system S unpredictable because of representation errors. In fact, most of the orbits obtained in computers are not even good approximations to the actual ones. Real number representations by computers fall under two headings: floating-point (or numerical) and algebraic (or symbolic). In the former case, given the finite amount of memory, only numbers with finite decimal representations are treated without (roundoff) errors. In the symbolic setting, exact representation extends to some alge braic irrationals, to some transcendentals, and to various algebraic combinations of these. Therefore, the development of a dynamical system inside a computer is prone to successive errors, which is usually fatal to predictions concerning chaotic systems. In this article, we look at the question of computational predictability more closely and show that unpredictability in chaotic systems is also related to the scheme in which the system is represented. For simplicity, we restrict our discussions to discrete-time dynamical systems (i.e., to systems S defined by maps / : V -> V for metric spaces V). The orbit of S corresponding to the initial condition x0 is defined as the sequence y : N -+ V given by y(n) ? xn ? fn(x0), where N = {0, 1, 2, ...}. The time variable of S is n, and V is its state space. The space V is usually a subset of M.d for some natural number d, the dimension of the system. The orbits of S may be open, eventually periodic, or periodic. We write (eventually) periodic when referring to either of the last two categories. We call S periodic if all its orbits are (eventually) periodic.

An exploration of hyperpower equations nx = ny

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown thathas an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as

Irrationals and the Fundamental Theorem of Arithmetic

Recent issues of the MONTHLY contain articles [1, 2] that discuss approaches to the of the square roots of nonsquare natural numbers using arguments designed to avoid any mention of unique factorization. The argument in [2] is incomplete but, as the article points out, the important objective when introducing a topic such as the existence of irrationals is to present arguments that the students find convincing and many students have difficulty with reasoning involving the unique factorization aspect of the Fundamental Theorem of Arithmetic. The following discussion is designed to give students some appreciation of the significance of the property of unique factorization by considering the irrationality of square roots in systems where this property does not hold. Ideally it should come immediately after the presentation of a plausibility argument such as that in [2] showing the students that square roots of primes are irrational. Given a subset A of the natural numbers that is closed under multiplication, we say an element c of A is A-composite provided c = ab where a and b are elements in A with a # 1 and b 0 1. An element c in A is called A-prime provided c 0 1 and c is not A-composite. For example, if A is the set of all multiples of 4, then 16, 48 and 80 are A-composites and 4, 40 and 100 are A-primes. We say a real number x is A-rational provided x can be expressed in the form a/b where a and b are elements of A. Note that each element c in A is A-rational since c can be expressed as C2/C. This is analogous to the fact that every natural number is rational. However, there are properties of the natural numbers that do not extend to any set A. For example, let A be the set of all multiples of 4 and let c 100. Since c is an A-prime, by analogy with the natural numbers its square root should not be A-rational, but Vc = a/b where a = 40 and b = 4. Such a situation is possible since in the set A the number a2 can be factored into A-primes as either a * a or c * b * b. There is nothing special about the multiples of 4 in the above example. A useful exercise is to have each student pick a subset of the natural numbers that is closed under multiplication and investigate the irrationality of roots in this system. This serves to illustrate the rarity of the unique factorization property and the crucial role played by this property in the study of algebraic irrationals.

On Minkowski’s singular function

The function f on the closed unit interval I to be considered here was introduced by Minkowski. It maps the algebraic irrationalities of degree at most 2 continuously to the rationals in I. If z = a/b and z' = c/d are (reduced) fractions, then we write z * z' for the mediant (a + c)/(b + d) and z z' for the arithmetic mean. Let Q denote the (dense) set of rationals in I with terminating dyadic expansion, i.e. the set of numbers x= n2'' (n = 0,.. .,2-1, v E N). Define a function g on Q inductively as follows: Let y' = 0, y= 1. Suppose the values yn = g(Xn) have been defined on level v. Then put y2+' = Yk (0 < k < 2W) and insert +1 =YV1 *Yk (1< k <2W) to obtain the images y+' = g(x+') on level v+ 1. The resulting function g is a strictly increasing continuous bijection from Q to the rationals. Note that g is constructed so as to have the property g(x) *g(x') = g(x x') whenever x, x' E Q, x' x = 21-, v E N. The inverse of g can be extended to the reals to yield a strictly increasing continuous distribution function f inducing a (singular) probability measure on I. We will be concerned with the regular and semiregular continued fraction expansions y=J +l 3+ [= la),...],l ,,... E N and y = a-i b= [a, b, ... .1, a, b, ... E N {1}. In the dyadic expansion of elements x E I we collect 0's and l's and write x = 0, -110O1 60, , letting a 1,/,-y,6, ,. . . denote string lengths (a, 3, -y, 6, E, ... E N). The following has been proved by different arguments (see [2] for references): (i) a+1;2i_j,'-y+2,26_1,E+2, ... f'(011 ,0,_1 16 0, ) (this is [2, (5)]); (ii) f is singular. Our purpose here is to present a simple unified approach to both properties by splitting up (i) into the following equalities, the second one making (ii) evident:

Simultaneous approximation of algebraic irrationalities

This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers θ1 and θ2 from a totally real algebraic number field, the constant γ12 can be explicitly calculated, this being the upper limit of the numbers c12 such that the inequality max (∥qθ1∥, ∥qθ2∥)⩽(qc12)−1/2 holds for infinitely many natural numbers q; likewise for the constant a12 such that the inequality ∥qθ1∥·∥qθ2∥ 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d \). is fixed.

Calculation of the regulator of a pure cubic field

A description is given of a modified version of Voronoi’s algorithm for obtaining the regulator of a pure cubic field Q ( D 3 ) Q(\sqrt [3]{D}) . This new algorithm has the advantage of executing relatively rapidly for large values of D. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers. The algorithm was implemented on a computer and run on all values of D ( ⩽ 10 5 ) D\;( \leqslant {10^5}) such that the class number of Q ( D 3 ) Q\;(\sqrt [3]{D}) is not divisible by 3. Several tables summarizing the results of this computation are also presented.

Continued fraction expansion of algebraic numbers

Every irrational x in (0,1) determines uniquely an infinite sequence of positive integers by means of its continued fraction expansion. Richtmyer in 1951 conjectured that algebraic irrationals may have a paucity of large partial denominators. The conjecture was studied in numerical tests. The numerical work yielded a very strange result for one value of x. The present paper reports further tests for this value. It was found that the previous anomalous value was accidental, and resulted from having stopped the expansion at 725 terms; other stopping points gave quite reasonable values. 4 tables. (RWR)

Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree $3$

It is not known whether or not the Jacobi-Perron Algorithm of a vector in ${R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied,

Errata: “Rational approximations of algebraic irrationals by means of a modified Jacobi-Perron algorithm,” vol. 32 (1965) pp. 161–176

Errata: “Rational approximations of algebraic irrationals by means of a modified Jacobi-Perron algorithm,” vol. 32 (1965) pp. 161–176

Rational approximations of algebraic irrationals by means of a modified Jacobi-Peron algorithm

Rational approximations of algebraic irrationals by means of a modified Jacobi-Peron algorithm