TY - JOUR
T1 - Polynomial averages and pointwise ergodic theorems on nilpotent groups
AU - Ionescu, Alexandru D.
AU - Magyar, Ákos
AU - Mirek, Mariusz
AU - Szarek, Tomasz Z.
N1 - Funding Information:
The first, second and third authors were supported in part by NSF Grants DMS-2007008 and DMS-1600840 and DMS-2154712 respectively. The third author was also partially supported by the Department of Mathematics at Rutgers University and by the National Science Centre in Poland, Grant Opus 2018/31/B/ST1/00204. The fourth author was partially supported by the National Science Centre of Poland, Grant Opus 2017/27/B/ST1/01623, the Juan de la Cierva Incorporación 2019, Grant Number IJC2019-039661-I, the Agencia Estatal de Investigación, Grant PID2020-113156GB-I00/AEI/10.13039/501100011033, the Basque Government through the BERC 2022-2025 program, and by the Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/3
Y1 - 2023/3
N2 - 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 Lp for 1 < p≤ ∞ and ρ-variational inequalities on L2 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.
AB - 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 Lp for 1 < p≤ ∞ and ρ-variational inequalities on L2 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.
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U2 - 10.1007/s00222-022-01159-0
DO - 10.1007/s00222-022-01159-0
M3 - Article
AN - SCOPUS:85138960219
SN - 0020-9910
VL - 231
SP - 1023
EP - 1140
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -