then so is sn. A van der Corput set is a set H of integers such that if for each h in H the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn. Metric theorems Metric theorems describe the behaviour of a parametrised sequence for almost all values of some parameter α: that is, for values of α not lying in some exceptional set of Lebesgue measure zero. For any sequence of distinct integers bn, the sequence (bnα) is equidistributed mod 1 for almost all values of α. The sequence (αn) is equidistributed mod 1 for almost all values of α > 1. It is not known whether the sequences (en) or (n) are equidistributed mod 1. However it is known that the sequence (αn) is not equidistributed mod 1 if α is a PV number. Well-distributed sequence A sequence (s1, s2, s3, ...) of real numbers is said to be well-distributed on [a, b] if for any subinterval [c, d] of [a, b] we have uniformly in k. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous. Sequences equidistributed with respect to an arbitrary measure For an arbitrary probability measure space , a sequence of points is said to be equidistributed with respect to if the mean of point measures converges weakly to : In any Borel probability measure on a separable, metrizable space, there exists an equidistributed sequence with respect to the measure; indeed, this follows immediately from the fact that such a space is standard. The general phenomenon of equidistribution comes up a lot for dynamical systems associated with Lie groups, for example in Margulis' solution to the Oppenheim conjecture. See also Equidistribution theorem Low-discrepancy sequence Erdős–Turán inequality Notes References Further reading External links Lecture notes by Charles Walkden with proof of Weyl's Criterion Diophantine approximation Dynamical systems Ergodic theory