منابع مشابه
Diophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملA note on Diophantine approximation
Given a set of nonnegative real numbers Λ= {λi}i=0, a Λ-polynomial (or Müntz polynomial) is a function of the form p(x)=ni=0 aizi (n∈N). We denote byΠ(Λ) the space of Λ-polynomials and byΠZ(Λ) := {p(x)=ni=0 aizi ∈Π(λ) : ai ∈ Z for all i≥ 0} the set of integral Λ-polynomials. Clearly, the sets ΠZ(Λ) are subgroups of infinite rank of Z[x] wheneverΛ⊂N, #Λ=∞ (by infinite rank, wemean that the real ...
متن کاملDiophantine Approximation in Banach Spaces
In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.
متن کاملDiophantine Approximation in Small Degree
Here, the exponent of q in the upper bound is optimal because, when ξ has bounded partial quotients, there is also a constant c > 0 such that |ξ − p/q| ≥ cq for all rational numbers p/q (see Chapter I of [14]). Define the height H(P ) of a polynomial P ∈ R[T ] as the largest absolute value of its coefficients, and the height H(α) of an algebraic number α as the height of its irreducible polynom...
متن کاملMultiplicative Diophantine approximation
In his paper, Dirichlet gives a complete proof for n = 1 and observes that this proof can be easily extended to arbitrary values of n. Good references on this topic are Chapter II of [52] and Cassels’ book [17]. There are in the literature many papers on various generalisations of the Dirichlet Theorem and on closely related problems. A typical question asks whether for a given set of mn real n...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1969
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-16-1-23-40