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Theoretical proof demonstrates pointwise convergence of ergodic averages on compact nilmanifolds, indicating polynomial orbits distribute evenly across sub-nilmanifolds.
We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.
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A. Leibman (2004) studied this question.
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