Equidistributed Sequence - Examples

Examples

  • The sequence of all multiples of an irrational α,
0, α, 2α, 3α, 4α, …

is uniformly distributed modulo 1: this is the equidistribution theorem.

  • More generally, if p is a polynomial with at least one irrational coefficient (other than the constant term) then the sequence p(n) is uniformly distributed modulo 1: this was proved by Weyl and is an application of van der Corput's difference theorem.
  • The sequence log(n) is not uniformly distributed modulo 1.
  • The sequence of all multiples of an irrational α by successive prime numbers,
2α, 3α, 5α, 7α, 11α, …

is equidistributed modulo 1. This is a famous theorem of analytic number theory, proved by I. M. Vinogradov in 1935.

  • The van der Corput sequence is equidistributed.

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