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 کاملTest sets of the knapsack problem and simultaneous diophantine approximation
Absact This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the co...
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 1917
ISSN: 0024-6115
DOI: 10.1112/plms/s2-16.1.294