@article{Banakh_Protasov_Slobodianiuk_2015, title={Scattered Subsets of Groups}, volume={67}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/1983}, abstractNote={We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset <em class="EmphasisTypeItalic ">A</em> of a group <em class="EmphasisTypeItalic ">G</em> is scattered if and only if <em class="EmphasisTypeItalic ">A</em> does not contain any piecewise shifted <em class="EmphasisTypeItalic ">IP</em> -subsets. For an amenable group <em class="EmphasisTypeItalic ">G</em> and a scattered subspace <em class="EmphasisTypeItalic ">A</em> of <em class="EmphasisTypeItalic ">G,</em> we show that <em class="EmphasisTypeItalic ">μ</em>(<em class="EmphasisTypeItalic ">A</em>) = 0 for each left invariant Banach measure <em class="EmphasisTypeItalic ">μ</em> on <em class="EmphasisTypeItalic ">G.</em> It is also shown that every infinite group can be split into ℵ<sub>0</sub&gt; scattered subsets.}, number={3}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Banakh, T. O. and Protasov, I. V. and Slobodianiuk, S. V.}, year={2015}, month={Mar.}, pages={304-312} }