Pointwise ergodic theorems along oscillating sequences
Abstract
We establish the first pointwise convergence result for nonconventional ergodic averages modeled along a large class of oscillating sequences, treating, for example, orbits of the form (⌊ ndsin(eπn)⌋)n∈ℕ, d∈ℕ. More specifically, for every probability space (X,ℬ,μ) equipped with an invertible measure-preserving transformation T: X→ X and every f∈ Lpμ(X) with p∈(1,∞], we prove that the limit limN→∞1/N∑n=1Nf(T⌊ P(n)ψ(αn)⌋x) exists for $μ$−a.e. $x∈ X$, where P is a non-constant polynomial with real coefficients, ψ is $1$-periodic and real-analytic, and α has irrationality exponent smaller than 1+√(deg(P)+1). On the one hand, the latter assumption is satisfied by a.e. α∈ℝ, and, on the other hand, our result covers any of the following concrete orbits ⌊ ncos(eπn)⌋, ⌊ n37sin(n)⌋, ⌊ √2n10esin(π(√[5]7+1) n)⌋, ⌊ ((log 2) n3+e n)eecos(π√2 n)⌋. The proof employs the so-called one-frequency circle method for estimating exponential sums with phases (ξ⌊ P(n)ψ(αn)⌋)1≤ n≤ N, but the major arc approximants of our exponential sums do not consist of their direct continuous counterparts. In the major arc regime we decouple the ψ-oscillatory part of the phase from the polynomial one, and approximate the exponential sum by ∫01(1/N∫0Ne2πiξP(t)ψ( s)dt)ds. This represents a novel phenomenon for problems of this kind, and, interestingly, our proof exploits the coefficient-free variational estimates of Radon-type operators.
Related
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- Pointwise ergodic theorems along sequences of intermediate growth
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- Pointwise convergence of noncommutative ergodic averages along the primes
- Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
BibTeX
@misc{daskalakis2026pointwise,
title = {{Pointwise ergodic theorems along oscillating sequences}},
author = {Leonidas Daskalakis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34301},
eprint = {2610.34301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34301},
}
RIS
TY - UNPB
TI - Pointwise ergodic theorems along oscillating sequences
AU - Leonidas Daskalakis
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34301
UR - https://arxiv.org/abs/2610.34301
AB - We establish the first pointwise convergence result for nonconventional ergodic averages modeled along a large class of oscillating sequences, treating, for example, orbits of the form (⌊ n^dsin(eπn)⌋)_n∈ℕ, d∈ℕ. More specifically, for every probability space (X,ℬ,μ) equipped with an invertible measure-preserving transformation T: X→ X and every f∈ L^p_μ(X) with p∈(1,∞], we prove that the limit lim_N→∞1/N∑_n=1^Nf(T^⌊ P(n)ψ(αn)⌋x) exists for $μ$−a.e. $x∈ X$, where P is a non-constant polynomial with real coefficients, ψ is $1$-periodic and real-analytic, and α has irrationality exponent smaller than 1+√(deg(P)+1). On the one hand, the latter assumption is satisfied by a.e. α∈ℝ, and, on the other hand, our result covers any of the following concrete orbits ⌊ ncos(eπn)⌋, ⌊ n^37sin(n)⌋, ⌊ √2n^10e^sin(π(√[5]7+1) n)⌋, ⌊ ((log 2) n^3+e n)e^e^cos(π√2 n)⌋. The proof employs the so-called one-frequency circle method for estimating exponential sums with phases (ξ⌊ P(n)ψ(αn)⌋)_1≤ n≤ N, but the major arc approximants of our exponential sums do not consist of their direct continuous counterparts. In the major arc regime we decouple the ψ-oscillatory part of the phase from the polynomial one, and approximate the exponential sum by ∫_0^1(1/N∫_0^Ne^2πiξP(t)ψ( s)dt)ds. This represents a novel phenomenon for problems of this kind, and, interestingly, our proof exploits the coefficient-free variational estimates of Radon-type operators.
ER -