Polynomial maps and polynomial sequences in groups

Hu, Ya-Qing

doi:10.48550/arxiv.2105.08000

Cited by 2 publications

(1 citation statement)

References 5 publications

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“…This is an interesting problem in its own right, corresponding to Waring-type problems on nilpotent groups, which may be interpreted as a question about solutions of suitable systems of Diophantine equations induced by the moment curve on G 0 . A qualitative variant of the Waring problem in the context of nilpotent groups was recently investigated in [30,31], see also the references given there.…”

Section: 41mentioning

confidence: 99%

Polynomial averages and pointwise ergodic theorems on nilpotent groups

et al. 2022

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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.Dedicated to the memory of Elias M. Stein our friend and advisor. Contents 1. Introduction 1 2. Notation and preliminaries 3. Ergodic theorems: Proof of Theorem 1.2 4. Maximal and variational estimates on G 0 : 2 theory 5. Minor arcs contributions:

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Section: 41mentioning

confidence: 99%

Polynomial averages and pointwise ergodic theorems on nilpotent groups

et al. 2022

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Polynomial averages and pointwise ergodic theorems on nilpotent groups

Ionescu¹,

Magyar²,

Mirek³

et al. 2021

Preprint

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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. Contents 1. Introduction 1 2. Notation and preliminaries 3. Ergodic theorems: Proof of Theorem 1.2 4. Maximal and variational estimates on G 0 : ℓ 2 theory 5. Minor arcs contributions:

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Polynomial maps and polynomial sequences in groups

Cited by 2 publications

References 5 publications

Polynomial averages and pointwise ergodic theorems on nilpotent groups

Polynomial averages and pointwise ergodic theorems on nilpotent groups

Polynomial averages and pointwise ergodic theorems on nilpotent groups

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