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Related Experiment Videos

Rational approximations to linear forms of exponentials and binomials.

G V Chudnovsky1

  • 1Department of Mathematics, Columbia University, New York, New York 10027.

Proceedings of the National Academy of Sciences of the United States of America
|May 1, 1983
PubMed
Summary

This study enhances Mahler's quantitative result by replacing exponentials with linear forms, yielding improved estimates for linear combinations of numbers. New irrationality exponents are established for transcendental numbers.

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Area of Science:

  • Number Theory
  • Diophantine Approximation

Background:

  • The Lindemann-Weierstrass theorem provides a foundation for transcendence theory.
  • Mahler established a quantitative estimate for linear combinations of exponentials of rational numbers.

Purpose of the Study:

  • To improve Mahler's quantitative estimate.
  • To generalize results using linear forms instead of simple exponentials.
  • To explore irrationality exponents of certain transcendental numbers.

Main Methods:

  • Replacing exponentials e(ri) with linearly independent linear forms L(i) = Sigma L(ij)e(sij).
  • Applying techniques from Diophantine approximation and transcendence theory.
  • Analyzing linear combinations of binomials (a/b)(ri).

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Main Results:

  • An improved quantitative estimate for linear combinations of linear forms of exponentials.
  • New results for linear combinations of binomials.
  • Demonstration of numbers like sinh 1 and sin 1 having an irrationality exponent of 2 + epsilon.

Conclusions:

  • The study provides a significant refinement of existing quantitative results in transcendence theory.
  • The generalization using linear forms broadens the applicability of these estimates.
  • The findings contribute to the understanding of the irrationality properties of transcendental numbers.