Ergodic Averages over Shrinking Sectors of ℤ[i]
Abstract
We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along Ω converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector--disk comparison in an explicit logarithmic shrinking range, with a power-saving error in log N, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and TΩ(n) x, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.
Related
- 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
- Noncommutative maximal inequalities for polynomial ergodic averages
BibTeX
@misc{burgin2026ergodic,
title = {{Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$}},
author = {Alex Burgin and Christina Giannitsi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25255},
eprint = {2609.25255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ergodic Averages over Shrinking Sectors of ℤ[i]
AU - Alex Burgin
AU - Christina Giannitsi
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25255
UR - https://arxiv.org/abs/2609.25255
AB - We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along Ω converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector--disk comparison in an explicit logarithmic shrinking range, with a power-saving error in log N, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and T^Ω(n) x, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.
ER -