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.

**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.