Diophantine approximation of a single irrational number
نویسندگان
چکیده
منابع مشابه
Diophantine Approximation and Analytic Number Theory
This conference dealt with two areas of Number Theory, “the queen of mathematics.” Diophantine approximation can be broadly described as the solvability in rational integers to various inequalities. The name comes from the later Greek mathematician Diophantus, who studied the solutions to certain equations. Though clearly a very old branch of mathematics, it remains a vibrant area of study to t...
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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...
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A partition of x > 0 of the form x = P i 2 i3i with distinct parts is called a double-base expansion of x. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest {2, 3}-integer, i.e., a number of the form 23, less than or equal to x. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Os...
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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 کاملUnsymmetrical Approximation of Irrational Numbers
1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitz's inequality that one side is strengthened and the other weakened. Notice that the result for r > l is ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1990
ISSN: 0022-314X
DOI: 10.1016/0022-314x(90)90102-w