Pointwise Ergodic Theorems Along ⌊ pc⌋ for Every c>1
Abstract
Let c>1 be noninteger, and let Λ denote the von Mangoldt function. The averages ANc f(x):=1/N∑n=1N Λ(n)f(T⌊ nc⌋x) satisfy the variational estimate |𝒱r(ANc f:N∈ℕ)|Lp(X) ≲c,p,r|f|Lp(X) for every 1<p<∞ and r>2, on every σ-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of Λ−1 with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the Möbius-weighted averages along ( ⌊ nc ⌋ )n ∈ ℕ throughout 1<p<∞.
Related
- Pointwise Ergodic Theorems Along Reiter Sequences on Locally Compact Groups
- Pointwise ergodic theorems along sequences of intermediate growth
- Pointwise ergodic theorems along oscillating sequences
- Pointwise convergence of noncommutative ergodic averages along the primes
- Pointwise convergence of bilinear ergodic averages for commuting transformations
BibTeX
@misc{fragkos2026pointwise,
title = {{Pointwise Ergodic Theorems Along $\lfloor p^c\rfloor$ for Every $c>1$}},
author = {Anastasios Fragkos},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02465},
eprint = {2611.02465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02465},
}
RIS
TY - UNPB
TI - Pointwise Ergodic Theorems Along ⌊ p^c⌋ for Every c>1
AU - Anastasios Fragkos
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02465
UR - https://arxiv.org/abs/2611.02465
AB - Let c>1 be noninteger, and let Λ denote the von Mangoldt function. The averages A_N^c f(x):=1/N∑_n=1^N Λ(n)f(T^⌊ n^c⌋x) satisfy the variational estimate |𝒱^r(A_N^c f:N∈ℕ)|_L^p(X) ≲_c,p,r|f|_L^p(X) for every 1<p<∞ and r>2, on every σ-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of Λ−1 with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the Möbius-weighted averages along ( ⌊ n^c ⌋ )_n ∈ ℕ throughout 1<p<∞.
ER -