Primitive Averages, Directional Expansivity, and Quantitative Twisted Recurrence for Ergodic $\mathbb{Z}^d$-Actions

Abstract

We prove two new results about probability preserving actions $T:\mathbb{Z}^d \curvearrowright (X,μ)$. First, for a function $f\in L^2(μ)$, we provide an explicit formula for the $L^2(μ)$-limit of the average \[\frac{1}{|Q_N^\mathcal{P}|}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where $\mathcal{P}\subset \mathbb{Z}^d$ is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is $1$, and $Q_N^\mathcal{P}= [-N,N]^d\cap \mathcal{P}$. Second, for a set $A\subset X$ with $μ(A)>0$, we provide a spectral condition under which the set of $\varepsilon$-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, μ\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to $1$. As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author (arXiv:2503.02501).