Effective irrationality measures and approximation by algebraic conjugates
Effective results for linear equations in two unknowns from a multiplicative division group
Effective results for restricted rational approximation to quadratic irrationals
Effective simultaneous approximation of complex numbers by conjugate algebraic integers
We study effectively the simultaneous approximation of n-1 different complex numbers by conjugate algebraic integers of degree n over ℤ(√-1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n-1 different complex numbers lie symmetrically about the real axis, then ℤ(√-1) can be replaced by ℤ. In Section 1 we prove an effective version of a Kronecker approximation theorem; we start with an idea of H....
Effective solution of the D(-1)-quadruple conjecture
Equations diophantiennes exponentielles
Exceptional units and cyclic resultants
Exceptional units and numbers of small Mahler measure.
Extensions of the Cugiani-Mahler theorem
In 1955, Roth established that if is an irrational number such that there are a positive real number and infinitely many rational numbers with and , then is transcendental. A few years later, Cugiani obtained the same conclusion with replaced by a function that decreases very slowly to zero, provided that the sequence of rational solutions to is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous...
Finiteness of odd perfect powers with four nonzero binary digits
We prove that there are only finitely many odd perfect powers in having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at -unit points (in a suitable -adic convergence), Roth’s...
Height Functions for Groups of S-units of Number Fields and Reductions Modulo Prime Ideals
A result on the orders of the reductions of an element of the group of S-units of a number field is obtained by investigating three height functions for groups of S-units of number fields which are analogous to local, global and canonical height functions for elliptic curves.
Homogeneous approximation in completions of A-fields of non-zero characteristic
How many primes can divide the values of a polynomial?
Integer Points and the Rank of Thue Elliptic Curves.
Integer solutions of a sequence of decomposable form inequalities
Introduction to Diophantine Approximation
In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].
Linear polynomials in numbers of bounded degree
Markoff Forms and Primitive Words.
Meilleures approximations d'une forme linéaire cubique
Mesures d'irrationalité de log 2.