Mahler measures in a cubic field

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.     

Counting algebraic numbers with large height II

We count algebraic numbers of fixed degree over a fixed algebraic number field. When the heights of the algebraic numbers are bounded above by a large parameter , we obtain asymptotic estimates for their cardinality as .

     

On the Trace of Algebraic Integers of Small Height

We give an upper bound for the modulus of the first non–zero trace among natural powers of an algebraic integer of small house. An upper bound for this power is obtained for the Pisot and Salem numbers. Although the house of these numbers is not at all small, similar bounds for the first non–zero trace are also established. Finally, we give an upper bound for the trace of an algebraic number with the Mahler measure bounded above by the square root of the degree.     

Certain Diophantine Properties of the Mahler Measure

It is proved that a polynomial in several Mahler measures with positive rational coefficients is equal to an integer if and only if all these Mahler measures are integers. An estimate for the distance between a metric Mahler measure and an integer is obtained. Finally, it is proved that the ratio of two distinct Mahler measures of algebraic units is irrational.     

On the Measure of a Nonreciprocal Algebraic Number

Let M() be the Mahler measure of an algebraic number and let G() be the modulus of the product of logarithms of absolute values of its conjugates. We prove that if is a nonreciprocal algebraic number of degree d 2 then M()² G()^1/d 1/2d. This estimate is sharp up to a constant. As a main tool for the proof we develop an idea of Cassels on an estimate for the resultant of and 1/. We give a number of immediate corollaries, e.g., some versions of Smyth's inequality for the Mahler measure of a nonreciprocal algebraic integer from below.     

On the spectrum of the Zhang-Zagier height

From recent work of Zhang and of Zagier, we know that their height is bounded away from 1 for every algebraic number different from . The study of the related spectrum is especially interesting, for it is linked to Lehmer's problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the so-called Zhang-Zagier inequality. To achieve this, we need some algebraic numbers of small height. So, in the third section, we describe an algorithm able to find them, and we give an algebraic number with height discovered in this way. This search up to degree 64 suggests that the spectrum of may have a limit point less than 1.292. We prove this fact in the fourth part.

     

Length of the Sum and Product of Algebraic Numbers

In the present paper, we consider products of lengths of algebraic numbers whose sum or product is a chosen algebraic number. These products are used to construct a new height function for algebraic numbers. With the help of this function, a metric on the set of all algebraic numbers, which induces the discrete topology, is introduced.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 854–860.Original Russian Text Copyright ©2005 by A. Dubickas, C. J. Smyth.     

Mahler型函数值的代数无关度量

In this paper,we give the algebraic independence measures for the values of Mahler type functions in complex number field and p-adic number field,respectively.     

满足Mahler型代数函数方程的函数值的p-adic超越度量

这篇文章给出了满足Mahler型代数函数方程的函数值的p-adic超越度量．     

On beta expansions for Pisot numbers

Given a number , the beta-transformation is defined for by (mod 1). The number is said to be a beta-number if the orbit is finite, hence eventually periodic. In this case is the root of a monic polynomial with integer coefficients called the characteristic polynomial of . If is the minimal polynomial of , then for some polynomial . It is the factor which concerns us here in case is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether must be cyclotomic in this case, particularly if . We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in (an infinite set), by a search up to degree in , to degree in , and to degree in . We find the smallest counterexample, the counterexample of smallest degree, examples where is nonreciprocal, and examples where is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to from above, and infinite sequences of with nonreciprocal which converge to from below and to the th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in . The Pisot numbers for which is cyclotomic are related to an interesting closed set of numbers introduced by Flatto, Lagarias and Poonen in connection with the zeta function of . Our examples show that the set of Pisot numbers is not a subset of .

     

The structure of the spectra of Pisot numbers

For q∈(1,2), Erdös, Joó and Komornik studied the set:

     

辅助函数范数的上界估计

Lehmer问题是当今数论中最经典的问题 ,很多数学家均对其进行了研究。其中辅助函数方法是最典型的方法 ,但大家一般是侧重于辅助函数的高 (heiht)的估计。而本文是从辅助函数的‖·‖2 范数的上界估计角度出发为寻求M(α)下界的更佳估计做准备     

Concentration at a radius for Hardy class functions

In this paper we establish the fundamental properties of concentration at a radius for functions in the classical Hardy space on the unit disk. For f(z) which is not identically zero and given r, 0<r<1, the concentration is defined via the ratio of the norm of f(rz) to the norm of f(z). Using the Mahler measure of f(z), we obtain information on the distribution of the zeros of f(z) in terms of the concentration ratio. In the last section of the paper, we examine the sharpness of concentration estimates for the Blaschke factor.     

Approximation to Real Numbers by Cubic Algebraic Integers I

In 1969, H. Davenport and W. M. Schmidt studied the problemof approximation to a real number by algebraic integers ofdegree at most 3. They did so, using geometry of numbers, byresorting to the dual problem of finding simultaneous approximationsto and ² by rational numbers with the same denominator. Inthis paper, we show that their measure of approximation forthe dual problem is optimal and that it is realized for a countableset of real numbers . We give several properties of these numbersincluding measures of approximation by rational numbers, byquadratic real numbers and by algebraic integers of degree atmost 3. 2000 Mathematics Subject Classification 11J04 (primary),11J13, 11J82 (secondary).     

Lehmer's problem for compact Abelian groups

We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.

     

Estimating Mahler measures using periodic points for the doubling map

The Mahler measure m(P)$m\left(P\right)$ of a polynomial P$P$ is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P)$m\left(P\right)$ which we illustrate with several examples.     

Fuglede-Kadison determinants and entropy for actions of discrete amenable groups

In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain -dynamical systems attached to Laurent polynomials , in terms of the (logarithmic) Mahler measure of . We extend the expansive case of their result to the noncommutative setting where gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede-Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

     

p-adic Transcendence and p-adic Transcendence Measures for the Values of Mahler Type Functions

We prove the p-adic transcendence and p-adic transcendence measures for the values of some Mahler type functions.     

Mahler型函数值的代数无关度量

利用初等的结式方法研究满足多项式形式的函数方程组的Mahler型函数的零点估计,给出了满足非线性函数方程组的Mahler型函数在代数点值的代效无关度量．