2 00 7 Less than continuum many translates of a compact nullset may cover any infinite profinite group Miklós

Less than continuum many translates of a compact nullset may cover any infinite profinite group Abstract We show that it is consistent with the axioms of set theory that every infinite profinite group G possesses a closed subset X of Haar measure zero such that less than continuum many translates of X cover G. This answers a question of Elekes and Tóth and by their work settles the problem for all infinite compact topological groups. Let G be a profinite group and let µ be a left Haar measure on G. If X ⊆ G is a measurable subset of measure zero, then trivially countably many G-translates of X can not cover G. Thus the continuum-hypothesis (CH) implies that one needs at least continuum many translates of X to do this. However, it turns out that if we do not assume CH, then, consistently with the axioms of set theory, the world may be different. Theorem 1 It is consistent with the axioms of set theory that for every infinite profinite group G there exists a closed subset X ⊆ G of Haar-measure 0 such that less than continuum many G-translates of X cover G. Note that our proof is not constructive in the sense that it uses a proba-bilistic argument. Theorem 1 answers a question of Elekes and Tóth [5, Question 3.10]. The first result of this type is due to Elekes and Steprans [4] who proved the analogue of Theorem 1 for the real line. For the background of covering the real line with less than continuum many nullsets see [2]. Recently Elekes and 1