Transfer-operator concentration for logarithmic seed complexity

Establish a transfer-operator concentration inequality for the Pisot beta-transformation that upgrades the unconditional O(N^2) public-seed bound for preserving all centroid-pair distances to a conditional O(log N) bound.

Background

The paper proves an unconditional public-seed existence result using a variance bound and a union bound over all centroid pairs, yielding a required projection dimension that scales quadratically with the number of centroids, O(N^2). It also states that a Bernstein-type concentration inequality would reduce this dependence to O(log N).

The authors identify the missing theoretical ingredient as a transfer-operator concentration inequality for the Pisot beta-transformation or its associated spectral-gap dynamics. Establishing such an inequality would provide exponentially small deviation probabilities for the squared-norm estimator and thereby justify the more efficient logarithmic dependence on the number of centroid pairs.