Diophantine approximation
Introduction
Diophantine approximation is a branch of number theory that deals with the approximation of real numbers by rational numbers. Named after the ancient Greek mathematician Diophantus, this field explores the precision with which irrational numbers can be approximated by fractions. The study is fundamental in understanding the distribution of rational numbers and has applications in various areas such as cryptography, dynamical systems, and algebraic geometry.
Historical Background
The origins of Diophantine approximation can be traced back to the works of Diophantus of Alexandria, who studied equations with integer solutions. However, the formal development of the field began in the 18th century with the work of Joseph-Louis Lagrange and Adrien-Marie Legendre. The subject gained significant traction with the contributions of Carl Friedrich Gauss, who introduced the concept of continued fractions, and later with Johann Dirichlet's approximation theorem.
Fundamental Concepts
Rational Approximations
A rational approximation of a real number \( \alpha \) is a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers, and \( q \neq 0 \). The quality of the approximation is measured by the absolute difference \( |\alpha - \frac{p}{q}| \).
Continued Fractions
Continued fractions provide a systematic way to approximate real numbers by rationals. Every real number can be expressed as an infinite continued fraction:
\[ \alpha = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}} \]
where \( a_0 \) is an integer and \( a_i \) (for \( i \geq 1 \)) are positive integers. The convergents of this continued fraction give the best rational approximations to \( \alpha \).
Dirichlet's Approximation Theorem
One of the foundational results in Diophantine approximation is Dirichlet's approximation theorem, which states that for any real number \( \alpha \) and any positive integer \( N \), there exist integers \( p \) and \( q \) such that:
\[ \left| \alpha - \frac{p}{q} \right| < \frac{1}{qN} \]
and \( 1 \leq q \leq N \).
Advanced Topics
Diophantine Equations
Diophantine equations are polynomial equations where the solutions are required to be integers. These equations are closely related to Diophantine approximation, as they often involve finding rational approximations to real numbers that satisfy certain conditions.
Transcendental Numbers
A number is called transcendental if it is not a root of any non-zero polynomial equation with rational coefficients. The approximation of transcendental numbers by rationals is a deep area of study. Liouville's theorem provides a criterion for identifying transcendental numbers based on their approximation properties.
Metric Diophantine Approximation
Metric Diophantine approximation deals with the measure-theoretic aspects of the approximation problem. It studies the distribution of rational approximations to real numbers and involves concepts such as Lebesgue measure and Hausdorff dimension.
Khintchine's Theorem
Khintchine's theorem is a central result in metric Diophantine approximation. It provides conditions under which almost all real numbers can be approximated by rationals with a given error bound. The theorem states that for a monotonic function \( \psi: \mathbb{N} \to \mathbb{R}^+ \), almost all real numbers \( \alpha \) satisfy:
\[ \left| \alpha - \frac{p}{q} \right| < \psi(q) \]
for infinitely many integers \( p \) and \( q \), if and only if the series \( \sum_{q=1}^{\infty} \psi(q) \) diverges.
Applications
Diophantine approximation has numerous applications in various fields:
Cryptography
In cryptography, Diophantine approximation techniques are used in algorithms for factoring large integers and in the construction of cryptographic protocols.
Dynamical Systems
In the study of dynamical systems, Diophantine approximation helps in understanding the behavior of orbits and the distribution of points on manifolds.
Algebraic Geometry
Diophantine approximation plays a role in algebraic geometry, particularly in the study of rational points on algebraic varieties and the distribution of these points.