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Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that |ktheta-h|<1/N. A slightly weaker form of the theorem states that ...

A phenomenon in which a system being forced at an irrational period undergoes rational, periodic motion which persists for a finite range of forcing values. It may occur for ...

The notation Q^_ denotes the algebraic closure of the rational numbers Q. This is equivalent to the set of algebraic numbers, sometimes denoted A.

For any real number r>=0, an irrational number alpha can be approximated by infinitely many rational fractions p/q in such a way that ...

An expression is said to be ambiguous (or poorly defined) if its definition does not assign it a unique interpretation or value. An expression which is not ambiguous is said ...

The symbol +/- is used to denote a quantity which should be both added and subtracted, as in a+/-b. The symbol can be used to denote a range of uncertainty, or to denote a ...

Consider the circle map. If K is nonzero, then the motion is periodic in some finite region surrounding each rational Omega. This execution of periodic motion in response to ...

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers ...

As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

The Littlewood conjecture states that for any two real numbers x,y in R, lim inf_(n->infty)n|nx-nint(nx)||ny-nint(ny)|=0 where nint(z) denotes the nearest integer function. ...