Glasner Property for (Semi-)Group Actions

Assume that a group G acts on a compact metric space X. We say that the action has Glasner property if for any infinite set A in X, and for any e >0 there exists g in G such that g A is e-dense in X. Examples of actions which have Glasner property include the action of N on R/Z by multiplication,the action of SL_n(Z) on (R/Z)_n, and others. Glasner introduced the property in 1979 and showed that the action of N on the torus R/Z has it. It follows from the compactness argument that if an action has Glasner property then for every e > 0, there exists k(e) such that for every set A in X with at least k(e) elements there exists g∈G such that g A is e-dense. In the case of the action of N on the torus, Alon and Peres proved that k(e)< e^(−2−d) for any d > 0. It is also known that the lower bound has to be at least quadratic, so they provided an almost optimal quantitative bound.

Andrew Rajchert

The University of Sydney

Andrew Rajchert is completing his third-year as an undergraduate student at the University of Sydney, pursuing a Bachelor of Science with majors in Mathematics and Data Science. He has participated in a number of research projects already, mainly focused probability and statistical algorithms. Andrew has also been a participant in Data61’s Vacation Scholarship, where he completed a research project in privacy and probabilistic data structures. Next year he will be completing his honours project at the University of Sydney, and plans on continuing in academia further.

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