Sumset phenomenon in locally compact topological groups (Q2665195)

In this article the authors prove Jin's Theorem for amenable non-compact locally compact topological groups using the ultrafilter method of M. Beiglböck. Let be \(G\) a locally compact non-compact topological group and let \(CB(G)\) denote the collection of all bounded complex valued continuous functions on \(G\) with the uniform norm. A function \(f\in CB(G)\) is a left norm continuous if \(g\mapsto f\circ \lambda_g:G\rightarrow CB(G)\) is norm continuous, where \(\lambda_g\) denotes the left translation by \(g\in G\). The collection of all left norm continuous functions on \(G\) is an \(m\)-admissible \(C^*\)-subalgebra of \(CB(G)\) and is denoted by \(Luc(G)\). For \(f\in Luc(G)\), \(f^{-1}(\{0\})\) is called a zero set. The collection of all compact subsets of \(G\) with positive Haar measure \(m\) is denoted by \(P_c(G)\). Let \(\mathcal{F}=\{F_n\}_{n\in D}\) be a net in \(P_c(G)\). The upper Banach density of \(A\subseteq G\) is defined by \(d^*_{\mathcal{F}}(A)=\sup\{\alpha:(\forall k\in D)(\exists n\geq k)(\exists g\in G)(m^*(A\cap F_n g)\geq \alpha m(F_n))\}\), where \(m^*\) is the outer measure of \(m\). The net \(\mathcal{F}\) is a (left) Følner net if and only if for each \(g\in G\), the net \(\left\{\frac{m(gF_n\Delta F_n)}{m(F_n)}\right\}_{n\in D}\) converges to 0. The group \(G\) is called amenable if there exists a sequence of compact subsets of \(G\) that is Følner. The following result plays an important role in this work, which is a weak version of Furstenberg's Principal Theorem. \begin{itemize} \item If \(A\) is a closed subset of \(G\) such that \(d^*_{\mathcal{F}}(A)>0\), where \(\mathcal{F}\) is a Fölner net in \(P_c(G)\), then there is a countably additive regular measure \(\mu\) on the set \(\mathcal{B}\) of Borel subsets of \(G^{Luc}\) such that \begin{itemize} \item[1.] \(\mu(\overline{A})=d^*_{\mathcal{F}}(A)\) (\(\overline{A}\) denotes the closure of \(A\) in \(G^{Luc}\)), \item[2.] for all closed subsets \(B\) of \(G\), \(\mu(\overline{B})\leq d^*_{\mathcal{F}}(B)\), \item[3.] for all \(B\in\mathcal{B}\) and all \(g\in G\), \(\mu(gB)=\mu(B)\), and \item[4.] \(\mu(G^{Luc})=1\). \end{itemize} \end{itemize} Some Ramsey Theoretic results have been obtained too. Finally the authors state Jin's Theorem. \begin{itemize} \item Let \(G\) be a \(\sigma\)-compact non-compact amenable topological group, then there exists left and right Følner sequences in \(P_c(G)\), and so it can define notions \(d^*_L\) and \(d^*_R\) as left and right Banach density, respectively. Let \(A\) and \(B\) be two zero subsets of \(G\) such that \(d^*_R(A)d^*_L(B)>0\), then \(B^{-1}A\) is (right) piecewise syndetic. (A subset \(A\) of \(G\) is called thick if and only if for every finite subset \(F\) of \(G\) there exists \(g\in G\) such that \(Fg\subseteq A\), and \(A\) is piecewise syndetic if there exists a finite subset \(H\) of \(G\) such that \(\bigcup_{h\in H}h^{-1}A\) is thick.) \end{itemize}