Infinitesimal containment and sparse factors of iid

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $Γ$: an action $(X,μ)$ is infinitesimally contained in $(Y,ν)$ if the statistics of the action of $Γ$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^Γ$ is infinitesimally contained in the left-regular action of $Γ$. For exact groups, this implies that sparse factor-of-iid subsets of $Γ$ are approximately hyperfinite. We use it to quantify a theorem of Chifan–Ioana on measured subrelations of the Bernoulli shift of an exact group. For the proof of infinitesimal containment we define \emph{entropy support maps}, which take a small subset $U$ of ${0,1}^I$ and assign weights to coordinates above every point of $U$, according to how ‘‘important’’ they are for the structure of the set.

💡 Research Summary

The paper introduces a new notion of comparison between probability‑measure‑preserving (p.m.p.) actions of a countable group Γ, called infinitesimal weak containment. While classical weak containment requires that the joint statistics of any finite family of measurable sets be approximated arbitrarily well, infinitesimal containment relaxes this requirement to small sets only: for any finite set F⊂Γ, ε>0 and any collection of measurable subsets A₁,…,A_k of X with sufficiently small measure (≤δ), one can find a scaling factor λ>0 and subsets B₁,…,B_k⊂Y such that the discrepancies |μ(A_i∩γA_j)−λ ν(B_i∩γB_j)| are bounded by ε·max_i μ(A_i). This captures the idea that the “local” dynamics on tiny pieces of X can be reproduced inside Y.

The central result (Theorem 1.6) shows that the Bernoulli shift (