648 research outputs found
Spectrum of the exponents of best rational approximation
Using the new theory of W. M. Schmidt and L. Summerer called parametric
geometry of numbers, we show that the going-up and going-down transference
inequalities of W. M. Schmidt and M. Laurent describe the full spectrum of the
exponents of best rational approximation to points in .Comment: 13 pages, 4 figures, minor corrections since version
Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer
In a series of recent papers, W. M. Schmidt and L. Summerer developed a new
theory by which they recover all major generic inequalities relating exponents
of Diophantine approximation to a point in , and find new ones.
Given a point in , they first show how most of its exponents of
Diophantine approximation can be computed in terms of the successive minima of
a parametric family of convex bodies attached to that point. Then they prove
that these successive minima can in turn be approximated by a certain class of
functions which they call -systems. In this way, they bring the
whole problem to the study of these functions. To complete the theory, one
would like to know if, conversely, given an -system, there exists a
point in whose associated family of convex bodies has successive
minima which approximate that function. In the present paper, we show that this
is true for a class of functions which they call regular systems.Comment: 11 pages, 1 figure, to appear in Journal de th\'eorie des nombres de
Bordeau
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