The Computational Complexity of Algebraic Numbers

Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

Algebraic Numbers with Integral Power Traces

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.

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.

Self-similarity in binary quasiperiodic sequences generated by a projection method is shown to exist when and only when it is associated with quadratic irrational numbers and the explicit self-similarity transformation for an arbitrary quadratic number is obtained. The self-similarity transformation is shown not to be reducible to the simplest form for a class of quadratic numbers.

Estimates for approximations to logarithms of rational numbers by rational numbers and quadratic irrationalities are established.

Let t be any complex number different from the poles of a Weierstrass elliptic function ℘(z), having algebraic invariants. Then we estimate from below the sum where α and β are algebraic numbers. The estimate is given in terms of the heights of α and β and the degree of the field Q(α, β), where Q is the field of rationals.

The number λq = 2cosπ/q, q ∈ N, q ≥ 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number and in some of these methods, the minimal polynomials of several algebraic numbers are used. Here we obtain the minimal polynomial of one of those numbers, cos(2π/n), over the field of rationals by means of the better known Chebycheff polynomials for odd q and give some of their properties. We calculated this minimal polynomial for n ∈ N by using the Maple language and classifying the numbers n ∈ N into different classes.

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.

A rational number is a number that can be written as a ratio of two numbers. This ratio is also called a fraction representation. A fraction representation includes two parts: the numerator (the number on top) and the denominator (the number on the bottom). The following is a fraction representation of a rational number:

The existence of the universal fractal spectra was demonstrated in a number of studies. The article considers a general approach demonstrating the occurrence of such spectra for various nature systems. It is based on two extremely general concepts: resonance and roughness of real physical systems. Application of the algorithm developed in the study causes the occurrence of two complementary fractals on the sets of rational and irrational numbers accordingly. Besides, it was shown that the power of equivalence classes of rational numbers is connected with well-known fact that resonance appears more easily for pairs of frequencies, which are small natural numbers. The results are used for the analysis of the spectrum of the periods of various nature systems.

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures u ( s ) , u ( r ) u(s),u(r) with r r being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if s = r s = r or s = 1 − r s = 1 - r . We also show that for each positive integer n &gt; 2 n{\text { &gt; 2}} , there are algebraic integers and fractionals of degree n n in the unit interval that are binomial numbers.

This paper formulates general conjectures relating the structure of the 2-classgroup C 2 ( d p ) {C_2}(dp) associated to Q ( d p ) {\mathbf {Q}}(\sqrt {dp} ) to the splitting of the ideal ( p ) in certain algebraic number fields. Here d ≢ 2 d\,\nequiv \,2 ( mod 4 ) \pmod 4 is a fixed integer and p varies over primes. The conjectures assert that there exists an algebraic number field Ω j ( d ) {\Omega _j}(d) such that the Artin symbol [ ( Ω j ( d ) / Q ) / ( p ) ] [({\Omega _j}(d)/{\mathbf {Q}})/(p)] determines the first j 2-invariants of the group C 2 ( d p ) {C_2}(dp) , i.e. it determines C 2 ( d p ) / C 2 ( d p ) 2 j {C_2}(dp)/{C_2}{(dp)^{{2^j}}} . These conjectures imply that the set of primes p for which C 2 ( d p ) {C_2}(dp) has a given set of 2-invariants has a natural density which is a rational number. Existing results prove the conjectures whenever j = 1 j = 1 or 2 and also for an infinite set of d with j = 3 j = 3 . The smallest open case is j = 3 j = 3 , d = − 21 d = - 21 . This paper presents evidence concerning these conjectures for d = − 4 d = - 4 , 8 and − 21 - 21 . Numerical evidence is given that Ω 3 ( − 21 ) {\Omega _3}( - 21) exists, and that natural densities which are rational numbers exist for the sets of primes with 2 j | h ( d p ) {2^j}|h(dp) for d = − 4 d = - 4 and 8, for 1 ⩽ j ⩽ 7 1 \leqslant j \leqslant 7 . A search for the hypothetical field Ω 4 ( − 4 ) {\Omega _4}( - 4) ruled out the simplest candidate fields: Ω 4 ( − 4 ) {\Omega _4}( - 4) is not a normal extension of Q of degree 16 ramifying only at (2).