11.3 Metric theory of Diophantine approximation

Metric theory of Diophantine approximation blends number theory and measure theory to study rational approximations of real numbers. It uses tools like Lebesgue measure and Hausdorff dimension to analyze the distribution and properties of well-approximable numbers.

Key theorems like Dirichlet's, Khintchine's, and Jarník's form the foundation of this field. Recent developments explore connections to dynamical systems, fractal geometry, and simultaneous approximation, deepening our understanding of number properties and distributions.

Metric Theory of Diophantine Approximation

Fundamental Concepts and Theorems

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• Metric theory of Diophantine approximation studies rational approximations to real numbers using measure-theoretic perspective
• Dirichlet's approximation theorem states for any real number α and positive integer N, integers p and q exist with 1 ≤ q ≤ N such that $|α - p/q| < 1/(qN)$
• Badly approximable numbers resist approximation by rationals better than a certain rate
• Khintchine's theorem classifies well-approximable numbers metrically, stating for almost all real numbers α, the inequality $|α - p/q| < ψ(q)/q$ has infinitely many solutions if and only if $\sum_{q=1}^{\infty} ψ(q)$ diverges
• Duffin-Schaeffer conjecture (now a theorem) generalizes Khintchine's result to broader approximating function classes
• Metric theory employs measure theory, ergodic theory, and dynamical systems to analyze approximable number distribution and properties

Applications and Advanced Concepts

• Well-approximable numbers allow better rational approximation than a certain rate, defined by approximation function ψ(q)
• Lebesgue measure of ψ-approximable number set determined by convergence or divergence of series $\sum_{q=1}^{\infty} ψ(q)$ (Khintchine's theorem)
• Hausdorff dimension provides refined measure for well-approximable number sets, especially for Lebesgue measure zero sets
• Jarník's theorem connects Hausdorff dimension of ψ-approximable number set to lower order of 1/ψ at infinity
• Mass Transference Principle transfers Lebesgue measure statements to Hausdorff measure statements in Diophantine approximation
• Hausdorff dimension study in Diophantine approximation links to fractal geometry and dynamical systems
• Recent developments explore Hausdorff dimension of sets defined by simultaneous Diophantine approximation and approximation in various metrics

Lebesgue Measure and Hausdorff Dimension

Measure Theory Concepts

• Lebesgue measure quantifies set size in n-dimensional Euclidean space
• Hausdorff dimension provides more refined measure for sets with Lebesgue measure zero
• Lebesgue measure of ψ-approximable numbers determined by series convergence $\sum_{q=1}^{\infty} ψ(q)$ (Khintchine's theorem)
• Hausdorff dimension of ψ-approximable numbers related to lower order of 1/ψ at infinity (Jarník's theorem)
• Mass Transference Principle allows transfer between Lebesgue and Hausdorff measure statements
• Connections exist between Hausdorff dimension, fractal geometry, and dynamical systems in Diophantine approximation

Applications to Well-Approximable Numbers

• Well-approximable numbers allow better rational approximation than specified by approximation function ψ(q)
• Set of ψ-approximable numbers has full Lebesgue measure if $\sum_{q=1}^{\infty} ψ(q)$ diverges (Khintchine's theorem)
• Hausdorff dimension provides deeper insight for Lebesgue measure zero sets of well-approximable numbers
• Jarník's theorem relates Hausdorff dimension to approximation quality, dim_H(W_ψ) = inf{s ≥ 0 : $\sum_{q=1}^{\infty} q^{s-1}ψ(q)^s$ converges}
• Mass Transference Principle extends results from Lebesgue to Hausdorff measure, enabling more precise set size analysis
• Recent research explores Hausdorff dimension of sets defined by simultaneous approximation (approximating multiple numbers simultaneously)
• Studies investigate Hausdorff dimension in different metrics (p-adic numbers, function fields)

Distribution of Algebraic and Transcendental Numbers

Algebraic Numbers

• Algebraic numbers are roots of polynomial equations with integer coefficients (√2, ∛3, (1+√5)/2)
• Liouville's theorem measures algebraic number irrationality, stating degree n algebraic numbers cannot be approximated better than $q^{-n}$ by rationals
• Thue-Siegel-Roth theorem improves Liouville's result, showing algebraic numbers cannot be approximated better than $q^{-2-ε}$ for any ε > 0
• Algebraic number set is countable, forming a measure zero subset of real numbers
• Schmidt's subspace theorem generalizes Thue-Siegel-Roth theorem to higher dimensions, impacting Diophantine approximation and transcendence theory
• Metric theory analyzes distribution of algebraic numbers of different degrees
• Recent studies explore Diophantine properties of specific algebraic number classes (totally real algebraic integers, Salem numbers)

Transcendental Numbers

• Transcendental numbers are real numbers that are not algebraic (π, e, 2^√2)
• Transcendental number set has full Lebesgue measure in real line, comprising "most" real numbers
• Metric theory provides tools to study approximation properties of transcendental numbers
• Mahler's classification divides transcendental numbers into classes based on their approximation by algebraic numbers
• Liouville numbers form a class of transcendental numbers with exceptionally good rational approximations
• Recent developments investigate Diophantine properties of specific transcendental number classes (periods, exponential functions of algebraic numbers)
• Connections between transcendental number theory and other areas (elliptic curves, modular forms) are actively researched

Metric Theory vs Continued Fractions

Continued Fraction Basics

• Continued fractions provide natural framework for studying rational approximations to real numbers
• Simple continued fraction expansion of real number α closely relates to its Diophantine approximation properties
• Convergents of continued fraction give best rational approximations in certain sense
• Partial quotient size in continued fraction determines rational approximation quality
• Lévy's theorem states for almost all real numbers, geometric mean of first n partial quotients tends to Khinchin's constant as n approaches infinity
• Bounded partial quotient numbers in continued fraction expansion coincide with badly approximable numbers (Lebesgue measure zero, full Hausdorff dimension)

Connections to Metric Theory

• Gauss map T(x) = {1/x} (fractional part of 1/x) on [0,1] encodes continued fraction algorithm
• Gauss map has deep connections to metric theory of Diophantine approximation
• Ergodic properties of Gauss map, including invariant Gauss measure, crucial for understanding continued fraction statistical properties
• Metric theory and continued fractions connect to hyperbolic geometry and modular group dynamics
• Continued fraction expansion statistics relate to Diophantine approximation properties (Khintchine-Levy theorem)
• Recent research explores connections between metric theory, continued fractions, and ergodic theory of homogeneous flows
• Studies investigate relationships between continued fraction expansions and Diophantine properties of specific number classes (quadratic irrationals, Sturmian sequences)