An ergodic theorem for nonsingular actions of the Heisenberg groups

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We show that there is a sequence of subsets of each discrete Heisenberg group for which the nonsingular ergodic theorem holds. The sequence depends only on the group; it works for any of its nonsingular actions. To do this, we use a metric which was recently shown by Le Donne and Rigot to have the Besicovitch covering property and then apply an adaptation of Hochman’s proof of the multiparameter nonsingular ergodic theorem.