Weighted ergodic averages along subpolynomials in Hardy fields and applications
Abstract
We establish new pointwise convergence results for weighted ergodic averages along sequences of the form (⌊ a(n) ⌋)n ∈ ℕ, where a(x) is a subpolynomial function in a Hardy field. For example, we establish pointwise convergence of logarithmic averages along sequences of the form (⌊ nk + logc n ⌋)n ∈ ℕ, where k ∈ ℕ ∪ {0} and c > 0. This result should be juxtaposed with the fact that either for k=0 or for k ≥ 2 and for sufficiently small c>0 (depending on k), the standard ergodic averages along these sequences fail to converge pointwise. We also obtain pointwise joint ergodicity results for multiple weighted ergodic averages along slow Hardy field functions. For example, it follows from our results that for c> 0 and for any f, g ∈ L∞ (λ), limN → ∞ 1/(log N ) ∑n=1N 1/n f(Tb⌊ logc n ⌋x) g(TG⌊ logc n ⌋ x) = ∫ f d λ· ∫ g d μG for almost every x ∈ [0,1], where Tb:[0,1] → [0,1] is the times-b map defined by $T_b x = bx \, \bmod \, 1 $ and TG:[0,1] → [0,1] is the Gauss map defined by TG(x) = 1/x bmod 1 for x ≠ 0 and TG (0) =0. Here λ is the Lebesgue measure on [0,1] and μG is the Gauss measure on [0,1] given by μG (A) = 1/( log 2) ∫A 1/(1+x) dx for any measurable set A ⊂ [0,1].
Related
- Weighted averages and applications to sets of multiple recurrence
- Multiple ergodic averages for commuting multiplicative actions
- Pointwise convergence of bilinear ergodic averages for commuting transformations
- Pointwise Convergence of Random Ergodic Averages at the L1 Endpoint
- Analysis of a zeroth-order multiscale sampler: strong convergence, convergence of ergodic averages and weak convergence
BibTeX
@misc{bergelson2026weighted,
title = {{Weighted ergodic averages along subpolynomials in Hardy fields and applications}},
author = {Vitaly Bergelson and Sovanlal Mondal and Younghwan Son},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14939},
eprint = {2609.14939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14939},
}
RIS
TY - UNPB
TI - Weighted ergodic averages along subpolynomials in Hardy fields and applications
AU - Vitaly Bergelson
AU - Sovanlal Mondal
AU - Younghwan Son
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14939
UR - https://arxiv.org/abs/2609.14939
AB - We establish new pointwise convergence results for weighted ergodic averages along sequences of the form (⌊ a(n) ⌋)_n ∈ ℕ, where a(x) is a subpolynomial function in a Hardy field. For example, we establish pointwise convergence of logarithmic averages along sequences of the form (⌊ n^k + log^c n ⌋)_n ∈ ℕ, where k ∈ ℕ ∪ {0} and c > 0. This result should be juxtaposed with the fact that either for k=0 or for k ≥ 2 and for sufficiently small c>0 (depending on k), the standard ergodic averages along these sequences fail to converge pointwise. We also obtain pointwise joint ergodicity results for multiple weighted ergodic averages along slow Hardy field functions. For example, it follows from our results that for c> 0 and for any f, g ∈ L^∞ (λ), lim_N → ∞ 1/(log N ) ∑_n=1^N 1/n f(T_b^⌊ log^c n ⌋x) g(T_G^⌊ log^c n ⌋ x) = ∫ f d λ· ∫ g d μ_G for almost every x ∈ [0,1], where T_b:[0,1] → [0,1] is the times-b map defined by $T_b x = bx \, \bmod \, 1 $ and T_G:[0,1] → [0,1] is the Gauss map defined by T_G(x) = 1/x bmod 1 for x ≠ 0 and T_G (0) =0. Here λ is the Lebesgue measure on [0,1] and μ_G is the Gauss measure on [0,1] given by μ_G (A) = 1/( log 2) ∫_A 1/(1+x) dx for any measurable set A ⊂ [0,1].
ER -