Introduction to Diophantine Approximation

By Yasushige Watase. Formalized Mathematics, 2015

411 citations recorded by the source corpus.

Abstract In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

Subfields: Algebra and Number Theory, Mathematical Physics, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/9652b485b4410bdf393de823b798412e3db824bf

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In Algebra and Number Theory

1. Serge Troubetzkoy
2. J. Schmeling
3. Yann Bugeaud
4. A. Fan
5. Larry Guth
6. Nicolas Chevallier
7. Y. Cheung
8. Samir Siksek

In Mathematical Physics

1. J. Schmeling
2. Serge Troubetzkoy
3. Yu Deng
4. Yann Bugeaud
5. A. Fan
6. Larry Guth
7. Nicolas Chevallier
8. Pierre Germain

In Theoretical Computer Science

1. Yann Bugeaud
2. Nicolas Chevallier
3. Y. Cheung

Citations in the corpus

• A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation (2013, background): “A counterpart in the irrational rotation, which is not chaotic, is the theorem of Markov-Minkowski concerning the Diophantine inequation ∥nα− y∥ < 1/( √ 5n) ([Cas], p.”
• Hausdorff dimension of singular vectors (2016, background): “The existence of singular vectors that do not lie in a rational subspace was proved by Khintchine for all dimensions ≥ 2.”
• Strichartz estimates for the Schrödinger equation on irrational tori (2017, background): “ue for all (β1,...,βd) outside of a null set (set with measure zero) of [1,2]d. Genericity will often be for us a consequence of a classical result on Diophantine approximation: it is well-known (see [8]) that, generically in (βi), there exists Csuch that |k1 +β2k2 +··· +βdkd| ≥ C 1 (|k1|+··· +|kd|)”
• Exponents of Diophantine approximation (2006, methodology): “3, Cassels [14], Chapter V, Theorem IV, or Schmidt [49], Chapter IV, Section 5.”
• Metric Diophantine Approximation : aspects of recent work (2016, neutral): “Happily, this is the case with the lim sup setW (ψ) of ψ-well approximable numbers [37, 36, 57].”