On Ergodic Sequences of Measures

Let Z Z be the group of integers and Z ¯ \bar Z its Bohr compactification. A sequence of probability measures { μ n , n = 1 , 2 , … } \{ {\mu _n},n = 1,2, \ldots \} defined on Z Z is said to be ergodic provided μ n {\mu _n} converges weakly to μ ¯ \bar \mu , the Haar measure on Z ¯ \bar Z . Let A n ⊂ Z , n = 1 , 2 , … {A_n} \subset Z,n = 1,2, \ldots and define μ n {\mu _n} by μ n ( B ) = | A n ∩ B | / | A n | {\mu _n}(B) = |{A_n} \cap B|/|{A_n}| where | B | |B| is the cardinality of B B . Then it is easy to show that if | A n ∩ A n + k | / | A n | → 1 |{A_n} \cap {A_n} + k|/|{A_n}| \to 1 for every k ∈ Z k \in Z , then μ n {\mu _n} is ergodic. Let 0 ≤ p k ≤ 1 0 \leq {p_k} \leq 1 . In this paper we construct (random) sequences { μ n } \{ {\mu _n}\} which are ergodic, and such that lim ( | A n ∩ A n + k | / | A n | ) = p k \lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k} , for every k ∈ Z k \in Z .