Effective Constructions in Transcendental Number Theory

Siegel’s Lemma is concerned with finding a “small” nontrivial integer solution of a large system of homogeneous linear equations with integer coefficients, where the number of variables substantially exceeds the number of equations (for example, n equations and N variables with N ≥ 2n), and “small” means small in the maximum norm. Siegel’s Lemma is a clever application of the Pigeonhole Principle, and it is a pure existence argument. The basically combinatorial Siegel’s Lemma is a key tool in transcendental number theory and diophantine approximation. David Masser (a leading expert in transcendental number theory) asked the question whether or not the Siegel’s Lemma is best possible. Here we prove that the so-called “Third Version of Siegel’s Lemma” is best possible apart from an absolute constant factor. In other words, we show that no other argument can beat the Pigeonhole Principle proof of Siegel’s Lemma (apart from an absolute constant factor). To prove this, we combine a concentration inequality (i.e., Fourier analysis) with combinatorics.

This volume consists of a collection of papers devoted primarily to transcendental number theory and diophantine approximations written by the author. Most of the materials included in this volume are English translations of the author's Russian manuscripts, extensively rewritten and brought entirely up to date. These papers and other papers included in this volume were available to specialists in manuscript form, but this is the first time that they have been collected and published.Though the earlier papers have been preserved in the form in which they were prepared initially, the volume is organized in such a way as to reflect recent progress and to allow readers to follow recent developments in the field. As an introductory guide to the volume, the author included an expanded and updated text of his invited address on his work on the theory of transcendental numbers to the 1978 International Congress of Mathematicians in Helsinki. The appendix contains a paper on the extremality of certain multidimensional manifolds prepared by A. I. Vinogradov and the author in 1976. Chudnovsky received a MacArthur Foundation Fellowship in 1981.

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

In this chapter, we will examine some of the emerging themes in the theory of transcendental numbers. The most fascinating is the “modular connection” linking it with the theory of modular forms. We have met a part of this connection in the earlier chapters. In this chapter, we will indicate some other relations.

The Siegel-Shidlovskii method is one of the main methods in the theory of transcendental numbers. By using this method, it is possible to establish the transcendency and algebraic independence of the values of entire functions belonging to a certain class (so-called E-functions) provided that these functions are algebraically independent over C(z). There are many examples of E-functions used in mathematics: exp(z), sinz, cosz, sinhz, coshz, Bessel functions, Kummer functions, incomplete gamma-function, generalized hypergeometric functions, and some other special functions. By using the Siegel-Shidlovskii method, it is possible to obtain the lower estimates for the moduli of polynomials from the values of E-functions. These estimates are called algebraic independence measures of the numbers. They serve as the quantitative characteristics of algebraic independence. The problem of obtaining the estimates of algebraic independence measures has long been dealt with by many researchers. The first estimates of measures for the values of exp(z) were obtained by E. Borel and K. Mahler, and for the values of Bessel functions such estimates were obtained by C. Siegel. Estimates of algebraic independence measures for the values of general E-functions were established by S. Lang, A.I. Galochkin, A.B. Shidlovskii, and Yu.V. Nesterenko. The constant appearing in the estimate of a measure is called effective if it can be expressed in terms of the parameters characterizing the considered functions and points at which their values are taken. An estimate of a measure is considered to be effective if it contains only effective constants. Generally speaking, the algebraic independence condition of functions is not sufficient for obtaining effective estimates of algebraic independence measures. Some additional conditions must be satisfied for this. The case in which the main totality of functions is algebraically dependent over C(z) is the most complicated one. In a number of the author’s works, the estimates of algebraic independence measures of the values of E-functions were improved. After 1995, no papers have been published on this topic, although it still remains of issue.

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = \( \sqrt 2 \). Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums.

Let G be a commutative algebraic group embedded in projective space and \(\Gamma \) a finitely generated subgroup of G. From these data we construct a chain of algebraic subgroups of G which is intimately related to obstructions to multiplicity or interpolation estimates used in transcendental number theory and algebraic independence. Let \(\gamma _1,\ldots ,\gamma _l\) denote a family of generators of \(\Gamma \) and, for any \(S>1\), let \(\Gamma (S)\) be the set of elements \(n_1\gamma _1+\cdots +n_l\gamma _l\) with integers \(n_j\) such that \(|n_j| < S\). Then this chain of subgroups controls, for large values of S, the distribution of \(\Gamma (S)\) with respect to algebraic subgroups of G. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all \(P \in H^0(\overline{G} ,\mathcal{O}(D))\) which vanish to at least some given order at all points of \(\Gamma (S)\). When D is very small this result reduces to a multiplicity estimate; when D is very large it is a kind of interpolation estimate.

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ ℝ is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond-Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of dimension $n$. Demailly has introduced a very interesting invariant --- the Seshadri constant $\epsilon(L,x)$ --- which in effect measures how positive $L$ is locally near a given point $x \in X$. For instance, Seshadri's criterion for ampleness may be phrased as stating that $L$ is ample if and only if there exists a positive number $e > 0$ such that $\epsilon(L,x) > e$ for all $x \in X$, and if $L$ is VERY ample, then $\epsilon(L,x) \ge 1$ for every $x$. We prove the somewhat surprising result that in each dimension $n$ there is a uniform lower bound on the Seshadri constant of an ample line bundle $L$ at a very general point of $X$. Specifically, $\epsilon(L,x) \ge (1/n) $ for all $x \in X$ outside the union of countably many proper subvarieties of $X$. Examples of Miranda show that there cannot exist a bound (independent of $X$ and $L$) that holds at every point. The proof draws inspiration from two sources: first, the arguments used to prove boundedness of Fano manifolds of Picard number one; and secondly some of the geometric ideas involving zero-estimates appearing in the work of Faltings and others on Diophantine approximation and transcendence theory. We give some elementary applications of the main theorem to adjoint and pluricanonical linear series.

For any irrational or transcendental number estimating of the quality of its approximation by rational fractions is one of the directions in the theory of Diophantine approximations. The quantitative characteristic of such approximation is called the measure (extent) of irrationality of the number. For almost a century and a half, scientists have developed various methods for evaluating the measure of irrationality and have obtained its values for a huge numberof irrational and transcendental numbers. Various approaches have been used to obtain the estimates and these approaches improved over time, leading to better estimates. The most commonly used method for obtaining such estimates is construction of linear forms with integer coefficients, which approximate a value, and studying of its asymptotic behavior. Approximating linear forms usually are constructed on the basis of continued fractions, Pad´e approximants,infinite series, and integrals. Methods for studying the asymptotics of such forms are currently quite standard, but the main problem is invention of a linear form with good approximating properties. The first estimates of the values of the arctangent function were obtained by M. Huttner in 1987 on the base of integral representation of the Gausss function. In 1993 A. Heimonen, T. Matala-Aho, K. Vaananen, using, like M. Huttner, Pad´e approximants for the Gaussianhypergeometric function, proved a general theorem for estimating of measures of irrationality of logarithms of rational numbers. Later, the same authors, using an approximating construction with Jacobi polynomials, obtained new estimates, in particular for the values of the function arctan 𝑥. Further research used various integral constructions, which allowed to get both general methods for arctan 𝑥 values and specialized methods for specific values. In the articles of E.B. Tomashevskaya, who in 2008 received a general estimate for the values of arctan 1/𝑛, 𝑛 ∈ N, was used a complex integral with the property of symmetry of integrand. This property played animportant role in obtaining the estimates, since it improved the asymptotic behavior of the coefficients of the linear form. Some integral constructions elaborated by other researchers alsohad different types of symmetry. In this article, we consider the main methods for estimating the values of the arctangent function, their features, research methods, and the best estimates at the moment.

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots of the polynomial and whose lengths are less than the given error bound, The algorithms also provide a simple means of determining the number of real roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 &lt; ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

Following the leads of H. Cohn and A. Schmidt we shall investigate geometric structures on hyperbolic Riemann surfaces for which Markoff-like theorems hold. The idea, which is similar to the approach taken by D. Sullivan in [21] is to look at the affinity of a geodesic for the noncompact end of a surface. More precisely, we consider the maximal depth a geodesic travels into a noncompact end. The spectrum of depths has the same structure as Markoff's spectrum with the correspondence given in terms of the geodesics length and topology. The upper discrete part of the spectrum is occupied by the simple closed geodesics and the lower limit value of the discrete spectrum is occupied by the geodesics that are limits of simple closed ones. The reason for this interaction between the geometry and the number theory becomes apparent when we transfer our attention to a Fuchsian group representing the hyperbolic surface. For example, consider the classical Modular group M6bz. The orbit of infinity under the action of M6bz is exactly the set of rational numbers. It is also the set of limit points which are fixed by parabolic transformations in the group. Each parabolic fixed point corresponds, in a sense, to the noncompact end on the quotient surface. As we shall see, the degree to which a number x is approximated by a rational number is directly related to the depth a geodesic with the endpoint x travels

Let f ( x ) f(x) be a polynomial with integer coefficients, and let \[ D ( f ) = g .c .d { f ( x ) : x ∈ Z } . D(f) = {\text {g}}{\text {.c}}{\text {.d}}\{ f(x):x \in {\mathbf {Z}}\}. \] It was conjectured by Bouniakowsky in 1857 that if f ( x ) f(x) is nonconstant and irreducible over Z {\mathbf {Z}} , then | f ( x ) | / D ( f ) |f(x)|/D(f) is prime for infinitely many integers x x . It is shown that there exist irreducible polynomials f ( x ) f(x) with D ( f ) = 1 D(f) = 1 such that the smallest integer x x for which | f ( x ) | |f(x)| is prime is large as a function of the degree of f f and the size of the coefficients of f f .

This textbook covers with the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense, but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. Nine main areas of the subject are treated: (i) divisibility and multiplicative functions; (ii) congruences and quadratic residues; (iii) the basics of algebraic numbers and sums of squares; (iv) continued fractions, diophantine approximations and transcendence; (v) quadratic forms; (vi) partitions; (vii) the prime numbers; (viii) diophantine equations; and (ix) elliptic curves (this part has been considerably extended for this edition). The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on fermat's last theorem is also breifly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.