Glossary

16.3 Diophantine approximation and transcendence theory

3 min read•july 30, 2024

and are powerful tools for understanding numbers. They help us figure out how close we can get to irrational numbers using fractions, and which numbers can't be solutions to polynomial equations.

These ideas are super useful for solving equations with whole number solutions. They also help us understand prime numbers better and even shed light on how certain mathematical systems behave over time.

Diophantine Approximation Concepts

Fundamentals and Theorems

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  • Diophantine approximation studies how well real numbers can be approximated by rational numbers
  • Measure "closeness" using absolute value and inequalities
  • theorem provides bounds for approximating irrational numbers (for any irrational α and positive integer N, integers p and q exist with 1 ≤ q ≤ N such that ∣α−p/q∣<1/(qN)|α - p/q| < 1/(qN))
  • offer a systematic method to find good rational approximations of real numbers
  • establishes a lower bound for approximating with rationals

Applications and Significance

  • Solve Diophantine equations (equations with integer solutions)
  • Study prime number distribution
  • Analyze dynamical systems behavior
  • Investigate rational point distribution on algebraic varieties

Practical Techniques

  • Use continued fraction expansions to approximate irrational numbers
    • Example: Approximate π using its continued fraction [3; 7, 15, 1, 292, ...]
  • Apply to determine irrationality measures of algebraic numbers
    • Example: Show that 2\sqrt{2} has irrationality measure 2
  • Employ the to bound integer solutions of Diophantine equations
    • Example: Bound solutions to x3−2y2=1x^3 - 2y^2 = 1

Diophantine Approximation vs Transcendence

Transcendence Theory Basics

  • Transcendence theory examines numbers not algebraic (not roots of non-zero polynomial equations with rational coefficients)
  • Connection established through study of rational approximations to algebraic and
  • Roth's theorem provides nearly optimal bound for rational approximations to algebraic numbers
  • Thue-Siegel-Roth theorem generalizes Roth's theorem, applicable in both fields

Advanced Connections

  • Baker's theory on linear forms in logarithms of algebraic numbers bridges the two fields
    • Provides tools for solving Diophantine equations
  • Mahler's classification of transcendental numbers (S-, T-, and ) relies on Diophantine approximation concepts
    • Example: are with infinite irrationality measure

Applications in Number Theory

  • Use transcendence results to study Diophantine equation solutions
    • Focus on equations involving exponential functions
  • Apply Diophantine approximation to analyze algebraic number properties
    • Example: Study the continued fraction expansion of ee to prove its irrationality

Key Results in Transcendence Theory

Fundamental Theorems

  • states algebraic independence of certain exponentials
    • If α1,...,αnα_1, ..., α_n are algebraic numbers linearly independent over rationals, then eα1,...,eαne^{α_1}, ..., e^{α_n} are algebraically independent over rationals
  • asserts transcendence of certain exponentials
    • If a and b are algebraic with a ≠ 0,1 and b irrational, then aba^b is transcendental
    • Example: 222^{\sqrt{2}} is transcendental
  • provides lower bound for absolute value of linear combinations of logarithms of algebraic numbers

Advanced Results and Conjectures

  • Schanuel's conjecture would unify many known transcendence results if proven true
  • Transcendence of π and e serve as fundamental results and introductions to transcendence methods
  • Six Exponentials Theorem and its generalizations offer powerful tools for proving transcendence and algebraic independence
    • Example: If x1,x2x_1, x_2 are linearly independent over rationals and y1,y2,y3y_1, y_2, y_3 are linearly independent over rationals, then at least one of exiyje^{x_iy_j} (i = 1,2; j = 1,2,3) is transcendental

Applications of Diophantine Approximation and Transcendence

Solving Specific Problems

  • Find rational approximations to irrational numbers using continued fractions
    • Example: Approximate 3\sqrt{3} using [1; 1, 2, 1, 2, 1, 2, ...]
  • Determine irrationality measures of algebraic numbers with Roth's theorem
    • Example: Show the irrationality measure of 23\sqrt[3]{2} is 3
  • Prove transcendence of specific numbers using Lindemann-Weierstrass theorem
    • Example: Prove eπe^π is transcendental

Advanced Problem-Solving Techniques

  • Solve exponential Diophantine equations using Baker's theory on linear forms in logarithms
    • Example: Find all integer solutions to 2x−3y=72^x - 3^y = 7
  • Study Diophantine equation solutions involving exponential functions using transcendence results
    • Example: Analyze the equation xy=yxx^y = y^x for integer solutions
  • Bound integer solutions to certain Diophantine equations with Thue-Siegel-Roth theorem
    • Example: Find an upper bound for solutions to x3−2y3=1x^3 - 2y^3 = 1

Key Terms to Review (21)

Algebraic numbers: Algebraic numbers are complex numbers that are roots of non-zero polynomial equations with integer coefficients. They include rational numbers, irrational numbers like the square root of 2, and even some complex numbers like the cube root of -1. Their significance lies in the study of Diophantine approximation and transcendence theory, which explore how these numbers can be approximated by rational numbers and their relationship to transcendental numbers.
Approximation constant: The approximation constant is a key concept in Diophantine approximation that quantifies how well a real number can be approximated by rational numbers. Specifically, it refers to a constant associated with a given real number that defines the best possible rate at which rational approximations converge to that number. This concept is crucial in understanding the boundaries of how close we can get to irrational or transcendental numbers using fractions.
Baker's Theorem: Baker's Theorem is a fundamental result in Diophantine approximation that establishes the existence of certain transcendental numbers. It asserts that if a number is algebraic, its approximation by rational numbers cannot be too close when the degree of the polynomial that defines it is taken into account. This theorem links to both Diophantine approximation and transcendence theory, emphasizing the limits of approximating algebraic numbers and highlighting the distinction between algebraic and transcendental numbers.
Continued fractions: Continued fractions are expressions of the form $$a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \cdots}}$$ where each $$a_i$$ is an integer. They serve as a powerful tool for approximating real numbers, particularly in the context of rational and irrational numbers, and have significant applications in Diophantine approximation and transcendence theory.
Diophantine approximation: Diophantine approximation is the study of how closely real numbers can be approximated by rational numbers. It focuses on finding good rational approximations to irrationals, which can lead to important results in number theory, particularly in understanding the properties of transcendental numbers.
Diophantine Property: The Diophantine property refers to the characteristic of a set of numbers being definable by a Diophantine equation, which is a polynomial equation where the solutions are sought in integers. This property highlights the connections between number theory and algebra, especially regarding which sets can be represented by such equations. In particular, understanding this property leads to insights in areas like Diophantine approximation and transcendence theory, revealing important limitations and capabilities of number sets.
Dirichlet's Approximation: Dirichlet's Approximation is a theorem in number theory that provides a way to approximate real numbers by rational numbers. It states that for any real number and any positive integer, there exists a rational number such that the absolute difference between the real number and the rational number is minimized, which is crucial for understanding how well real numbers can be approximated by rationals. This theorem plays a significant role in Diophantine approximation and has implications in transcendence theory.
Gelfond-Schneider Theorem: The Gelfond-Schneider Theorem states that if $a$ is an algebraic number (not equal to 0 or 1) and $b$ is a non-zero algebraic number, then the number $a^b$ is a transcendental number. This theorem has significant implications in both Diophantine approximation and transcendence theory, as it provides a connection between algebraic numbers and the broader category of transcendental numbers.
Khintchine's Theorem: Khintchine's Theorem is a fundamental result in Diophantine approximation that describes the behavior of the rational approximations of real numbers. Specifically, it establishes that for almost all real numbers, the size of their best rational approximations is governed by the continued fraction expansion of those numbers, leading to a specific asymptotic formula for the approximation errors.
Lindemann-Weierstrass Theorem: The Lindemann-Weierstrass Theorem states that if you have a set of distinct algebraic numbers, then the exponential functions of these numbers are linearly independent over the rational numbers. This theorem is crucial in the study of transcendental numbers, illustrating that certain combinations of algebraic numbers lead to transcendental results and playing a key role in both Diophantine approximation and transcendence theory.
Liouville numbers: Liouville numbers are a special class of real numbers that can be approximated by rational numbers with extraordinary accuracy. Specifically, a number is called a Liouville number if for every positive integer n, there exist infinitely many pairs of integers (p, q) such that $$|x - \frac{p}{q}| < \frac{1}{q^{n}}$$. This concept is pivotal in understanding Diophantine approximation and provides crucial insights into transcendence theory, as Liouville numbers are examples of transcendental numbers, which are not roots of any non-zero polynomial equation with integer coefficients.
Liouville's Theorem: Liouville's Theorem states that if a function is entire (holomorphic on the entire complex plane) and bounded, then it must be a constant function. This theorem is significant in understanding properties of complex functions and has strong implications in the fields of Diophantine approximation and transcendence theory, as it illustrates the limitations of non-constant entire functions.
Mahatma's Theorem: Mahatma's Theorem is a result in number theory that deals with the approximation of algebraic numbers by rational numbers. It plays a crucial role in understanding Diophantine approximations, providing insights into the properties and behaviors of algebraic numbers in relation to rational numbers. This theorem connects with transcendence theory by offering tools to analyze whether certain numbers can be expressed as roots of polynomials with integer coefficients.
Number theory: Number theory is a branch of mathematics focused on the properties and relationships of integers. It examines various aspects, including divisibility, prime numbers, and the solutions of equations involving whole numbers. This area of study serves as a foundation for other mathematical concepts, influencing areas like the distribution of primes and the approximation of rational numbers.
Roth's Theorem: Roth's Theorem is a significant result in number theory that states if a real number is algebraic and not rational, then it cannot be approximated too closely by rational numbers. Specifically, it provides a bound on how well an algebraic number can be approximated by rational numbers, asserting that the set of such approximations is limited. This theorem connects deeply with Diophantine approximation, as it addresses how closely algebraic numbers can be approximated by simpler forms, such as fractions.
S-numbers: s-numbers are a type of real or complex number that can be expressed as a limit of rational numbers and play a significant role in Diophantine approximation. They are defined in relation to the approximation of algebraic numbers by rational numbers, and they help establish the boundaries for the approximation of these numbers by identifying certain properties related to their continued fractions and growth rates.
T-numbers: T-numbers are specific algebraic integers that can be expressed as solutions to certain types of equations, particularly in the context of Diophantine approximation and transcendence theory. These numbers arise in the study of approximating real numbers by rational numbers and their connections to transcendental numbers, providing insight into the nature of number theory and the limitations of numerical approximation.
Thue-Siegel-Roth Theorem: The Thue-Siegel-Roth Theorem is a fundamental result in Diophantine approximation that establishes bounds on how closely algebraic numbers can be approximated by rational numbers. It asserts that if $ heta$ is an algebraic number of degree at least 2, then for any $eta > 0$, there are only finitely many rational numbers $p/q$ such that the distance between $ heta$ and $p/q$ is less than $1/q^{2+eta}$. This theorem bridges the gap between number theory and transcendence theory by providing insights into the limits of rational approximations to algebraic numbers.
Transcendence theory: Transcendence theory is a branch of mathematics focused on the study of transcendental numbers and their properties, particularly their non-algebraic nature. It examines how these numbers cannot be roots of any non-zero polynomial equation with rational coefficients, distinguishing them from algebraic numbers. This theory plays a crucial role in understanding Diophantine approximation, which involves approximating real numbers by rational numbers, and in establishing the existence of transcendental numbers.
Transcendental Numbers: Transcendental numbers are real or complex numbers that are not roots of any non-zero polynomial equation with rational coefficients. This means they cannot be expressed as solutions to algebraic equations, distinguishing them from algebraic numbers. Transcendental numbers have deep implications in various areas of mathematics, particularly in field extensions and transcendence theory, where they help us understand the structure and properties of different number systems.
U-numbers: U-numbers are a class of algebraic numbers characterized by their unique properties in the context of Diophantine approximation and transcendence theory. These numbers can be understood as solutions to certain polynomial equations, which exhibit interesting relationships with algebraic integers and transcendental numbers, particularly in how they relate to approximating irrational numbers by rationals.
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