Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on σ -finite measure spaces. We also establish corresponding maximal inequalities on L p for 1 < p ≤ ∞ and ρ -variational ineq...

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Bibliographic Details
Published in:Inventiones mathematicae 2023-03, Vol.231 (3), p.1023-1140
Main Authors: Ionescu, Alexandru D., Magyar, Ákos, Mirek, Mariusz, Szarek, Tomasz Z.
Format: Article
Language:English
Subjects:
Ergodic processes
Fourier transforms
Grants
Inequalities
Mathematics
Mathematics and Statistics
Orthogonality
Polynomials
Theorems
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Summary:We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on σ -finite measure spaces. We also establish corresponding maximal inequalities on L p for 1 < p ≤ ∞ and ρ -variational inequalities on L 2 for 2 < ρ < ∞ . This gives an affirmative answer to the Furstenberg–Bergelson–Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-022-01159-0