Examples
- The sequence of all multiples of an irrational α,
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- 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.
Read more about this topic: Equidistributed Sequence
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)