For every real alpha and rational gamma, averaging with powers T^{floor(alpha n)} and T^{floor(gamma n)} converges almost everywhere, and the return-times analogue holds for aperiodic systems.
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A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times
For every real alpha and rational gamma, averaging with powers T^{floor(alpha n)} and T^{floor(gamma n)} converges almost everywhere, and the return-times analogue holds for aperiodic systems.