3,566 research outputs found
Report on some recent advances in Diophantine approximation
A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.Comment: to be published by Springer Verlag, Special volume in honor of Serge
Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and
John Tat
Elliptic curves and continued fractions
We detail the continued fraction expansion of the square root of the general
monic quartic polynomial, noting that each line of the expansion corresponds to
addition of the divisor at infinity. We analyse the data yielded by the general
expansion. In that way we obtain `elliptic sequences' satisfying Somos
relations. I mention several new results on such sequences. The paper includes
a detailed `reminder exposition' on continued fractions of quadratic
irrationals in function fields.Comment: v1; final -- I hop
Continued fraction for formal laurent series and the lattice structure of sequences
Besides equidistribution properties and statistical independence the lattice profile, a generalized version of Marsaglia's lattice test, provides another quality measure for pseudorandom sequences over a (finite) field. It turned out that the lattice profile is closely related with the linear complexity profile. In this article we give a survey of several features of the linear complexity profile and the lattice profile, and we utilize relationships to completely describe the lattice profile of a sequence over a finite field in terms of the continued fraction expansion of its generating function. Finally we describe and construct sequences with a certain lattice profile, and introduce a further complexity measure
Hankel determinants, Pad\'e approximations, and irrationality exponents
The irrationality exponent of an irrational number , which measures the
approximation rate of by rationals, is in general extremely difficult to
compute explicitly, unless we know the continued fraction expansion of .
Results obtained so far are rather fragmentary, and often treated case by case.
In this work, we shall unify all the known results on the subject by showing
that the irrationality exponents of large classes of automatic numbers and
Mahler numbers (which are transcendental) are exactly equal to . Our classes
contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the
Fermat numbers, the regular paperfolding numbers, which have been previously
considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new
classes such as the Stern numbers and so on. Among other ingredients, our
proofs use results on Hankel determinants obtained recently by Han.Comment: International Mathematics Research Notices 201
Ultrametric Logarithm Laws I
We announce ultrametric analogues of the results of Kleinbock-Margulis for
shrinking target properties of semisimple group actions on symmetric spaces.
The main applications are S-arithmetic Diophantine approximation results and
logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on
trees.Comment: This announcement has been completely revised to reflect many new
developments. Please direct all references to this NEW announcement. It is
now co-authored work. Submitted to Discrete and Continuous Dynamical System
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