Lp-Khinchin sequences and transference principles
Abstract
A sequence (λn)n≥1 of distinct positive integers is called an Lp-Khinchin sequence, where 1≤ p≤∞, if $ n^{-1}\sum_{k=1}^n f(λ_k x) \longrightarrow \int_{\mathbb T}f\,d\mathfrak m $ a.e. x∈𝕋 for every f∈ Lp(𝕋). When 1≤ p<∞, we characterize the Lp-Khinchin sequences by a weak (p,p) maximal inequality for the associated averaging operators. We also establish a transference principle reducing the weak (p,p) maximal inequality to weak (p,p) maximal inequalities for three model systems: translation along a lattice path in the infinite direct sum of the semigroup ℕ0, translation along a lattice path in the infinite direct sum of the group ℤ and translation on the circle 𝕋. As consequences, we prove that (2n+1)n≥1, the Fibonacci sequence, sequences of distinct primes and their partial products are \emph{not} Lp-Khinchin for any 1≤ p<∞. We study the special class of multiplicative sequences generated by infinite words and we show that balanced infinite words generate L1-Khinchin sequences, while typical random multiplicative sequences over at least two coprime integers are almost surely not Lp-Khinchin for any 1≤ p<∞. We also obtain a {Conze-type principle} for countably generated semigroup actions on a probability space.
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BibTeX
@misc{fan2026lpkhinchin,
title = {{$L^p$-Khinchin sequences and transference principles}},
author = {Aihua Fan and Shilei Fan and Tongjin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26152},
eprint = {2610.26152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26152},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - L^p-Khinchin sequences and transference principles
AU - Aihua Fan
AU - Shilei Fan
AU - Tongjin Li
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26152
UR - https://arxiv.org/abs/2610.26152
AB - A sequence (λ_n)_n≥1 of distinct positive integers is called an L^p-Khinchin sequence, where 1≤ p≤∞, if $ n^{-1}\sum_{k=1}^n f(λ_k x) \longrightarrow \int_{\mathbb T}f\,d\mathfrak m $ a.e. x∈𝕋 for every f∈ L^p(𝕋). When 1≤ p<∞, we characterize the L^p-Khinchin sequences by a weak (p,p) maximal inequality for the associated averaging operators. We also establish a transference principle reducing the weak (p,p) maximal inequality to weak (p,p) maximal inequalities for three model systems: translation along a lattice path in the infinite direct sum of the semigroup ℕ_0, translation along a lattice path in the infinite direct sum of the group ℤ and translation on the circle 𝕋. As consequences, we prove that (2^n+1)_n≥1, the Fibonacci sequence, sequences of distinct primes and their partial products are \emph{not} L^p-Khinchin for any 1≤ p<∞. We study the special class of multiplicative sequences generated by infinite words and we show that balanced infinite words generate L^1-Khinchin sequences, while typical random multiplicative sequences over at least two coprime integers are almost surely not L^p-Khinchin for any 1≤ p<∞. We also obtain a {Conze-type principle} for countably generated semigroup actions on a probability space.
ER -