Number theory
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Approximation
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Continued fraction
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. The first problem was to know how well...
Low-discrepancy sequence
Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy.Roughly speaking, the discrepancy of a sequence i...
Low-discrepancy sequence - Wikipedia
Hurwitz's theorem (number theory)
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relat...
Kronecker's theorem
In mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.
This is a theorem stating that a non-constant polynomial in a field, p(x) ∈ F[x], has a...
Davenport–Schmidt theorem
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it ...
Discrepancy of hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory.
In the classical setting, we aim at partitioning the vertices of a hypergraph into two classes in such a way that ideally each hyperedg...
Littlewood conjecture
In mathematics, the Littlewood conjecture is an open problem (as of 2011) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β...
Restricted partial quotients
In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotie...
Restricted partial quotients - Wikipedia
Quasi-Monte Carlo method
In numerical analysis, quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random ...
Van der Corput sequence
A van der Corput sequence is the simplest one dimensional low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed b...
Equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequence is uniformly distributed on the circle , when a is an irrational number. It is a special case of the ergodic theorem wh...
Markov number
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equationstudied by Andrey Markoff (1879, 1880).The first few Markov numb...
Markov number - Wikipedia
Equidistributed sequence
In mathematics, a sequence {s1, s2, s3, …} of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of...
Constructions of low-discrepancy sequences
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy.Roughly speaking, the discrepancy of a sequence i...
Harmonious set
In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual ...
Duffin–Schaeffer conjecture
The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if is a real-valued function taking on positive...
Lagrange number
In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.
Weyl's inequality
In mathematics, there are at least two results known as "Weyl's inequality".
In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q co...
Markov spectrum
In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in the theory of diophantine approximation, and containing all the real numbers larger than Fr...
Oppenheim conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the ...
Schneider–Lang theorem
In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Her...
Quasi-Monte Carlo methods in finance
High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold . If the integral is of dimension the...
Quasi-Monte Carlo methods in finance - Wikipedia
Liouville number
In number theory, a Liouville number is an irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such thatA Liouville number can thu...
Liouville number - Wikipedia
Siegel's lemma
In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence o...
Sobol sequence
Sobol sequences (also called LPτ sequences or (t, s) sequences in base 2) are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician I....
Halton sequence
In statistics, Halton sequences are sequences used to generate points in space for numerical methods such as Monte Carlo simulations. Although these sequences are deterministic they are of low discr...
Halton sequence - Wikipedia
Beatty sequence
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiplesof a positive irrational number. Beatty sequences are ...
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number α and any positive integer N, there exists integers p...
Discrepancy theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the ...
Discrepancy theory - Wikipedia
Subspace theorem
In mathematics, the subspace theorem is a result obtained by Wolfgang M. Schmidt (1972). It states that if L1,...,Ln are linearly independent linear forms in n variables with algebraic coeffi...
Faltings' product theorem
In arithmetic geometry, the Faltings product theorem gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introd...