Dirichlet's approximation theorem

For every real number α and every positive integer N, there are integers p and q with 1 ≤ q ≤ N such that |α − p/q| < 1/(qN). Consequently, irrational numbers have rational approximations with arbitrarily large denominators and error below 1/q².

Pigeonhole principle: If more objects are distributed among fewer containers, some container holds at least two objects. Partitioning fractional parts into intervals produces two whose difference yields the required approximation.
Irrational number: A real number that cannot be expressed as a ratio of two integers. For irrationals, the theorem yields approximations with unbounded denominators and error below the reciprocal square.
Diophantine approximation: The study of integer solutions to equations and of how closely numbers can be approximated by rationals. Dirichlet's theorem is a foundational universal result in this broader area.
Equidistribution theorem: A result asserting that a sequence is uniformly distributed when its terms visit every region with the expected frequency. Diophantine bounds help control how rapidly certain sequences become uniformly distributed.
Legendre's theorem: A theorem stating that a reduced fraction approximating a real number within 1/(2q²) must be a continued-fraction convergent. It identifies when a sufficiently strong approximation must come from a continued-fraction convergent.
Fractional part: For a real number x, its fractional part is x minus its greatest integer not exceeding x. The proof compares fractional parts of multiples of the number being approximated.
Badly approximable number: An irrational number whose rational approximations cannot have errors smaller than a fixed positive constant times the reciprocal denominator squared. Such numbers show that the theorem's exponent is sharp, though its constant can be improved for particular numbers.
Denominator: The lower number in a fraction, indicating how many equal parts make up each unit fraction. The theorem controls approximation error in terms of the rational approximation's denominator.
Kronecker's theorem: A theorem characterizing when sequences generated by repeated additions on a torus are dense or uniformly distributed. Rational approximation governs whether rotations by an irrational angle explore the circle densely.
Dirichlet's theorem on primes in arithmetic progressions: A theorem asserting infinitely many primes in every arithmetic progression whose initial term and common difference are coprime. It shares Dirichlet's name but concerns primes rather than rational approximation.

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