Definition
Let be an infinite, strictly increasing sequence of positive integers. Then, given an integer q, this sequence is said to be ergodic mod q if, for all integers, one has
where
and card is the count (the number of elements) of a set, so that is the number of elements in the sequence A that are less than or equal to t, and
so is the number of elements in the sequence A, less than t, that are equivalent to k modulo q. That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod q as the sequence is taken to infinity.
An equivalent definition is that the sum
vanish for every integer k with .
If a sequence is ergodic for all q, then it is sometimes said to be ergodic for periodic systems.
Read more about this topic: Ergodic Sequence
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