57 research outputs found
On Salem numbers, expansive polynomials and Stieltjes continued fractions
A converse method to the Construction of Salem (1945) of convergent families
of Salem numbers is investigated in terms of an association between Salem
polynomials and Hurwitz quotients via expansive polynomials of small Mahler
measure. This association makes use of Bertin-Boyd's Theorem A (1995) of
interlacing of conjugates on the unit circle; in this context, a Salem number
is produced and coded by an m-tuple of positive rational numbers
characterizing the (SITZ) Stieltjes continued fraction of the corresponding
Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over
a Salem polynomial having simple roots, not cancelling at , coming from
monic expansive polynomials of constant term equal to their Mahler measure, has
a semigroup structure. The sets of corresponding generalized Garsia numbers
inherit this semi-group structure.Comment: 35 pages, Journal de Th{\'e}orie des nombres de Bordeaux, Soumissio
On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions
International audienceLet be an integer and denote by the real root in of the trinomial . The sequence of Perron numbers tends to 1. We prove thatthe Conjecture of Lehmer is true for by the direct method of Poincar\'e asymptoticexpansions (divergent formal series of functions) of the roots , of lying in , as a function of only. This method, not yet applied to Lehmer’s problem up to theknowledge of the author, is successfully introduced here. It first gives the asymptotic expansionof the Mahler measures of the trinomials as a function of only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisotnumber. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. Bythis method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for, with a minoration of the house \house\{\theta_{n}^{−1}\}= \theta_{n}^{−1} , and a minoration of the Mahler measure better than Dobrowolski’s one for . The angular regularity of the roots of , near the unitcircle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu,Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context ofthe Erd\H{o}s-Tur\'an-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions
Open Diophantine Problems
We collect a number of open questions concerning Diophantine equations,
Diophantine Approximation and transcendental numbers. Revised version:
corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1
(2004) dedicated to Pierre Cartie
Diophantische Approximationen
This number theoretic conference was focused on a broad variety of subjects in (or closely related to) Diophantine approximation, including the following: metric Diophantine approximation, Mahler’s method in transcendence, geometry of numbers, theory of heights, arithmetic dynamics, function fields arithmetic
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