Upper and lower frequently universal series (Q2845576)

The author introduces the notion of upper and lower frequently universal sequences and shows that ``most'' of the universal approximations are obtained by sets of indices which have upper density 1 and lower density 0. More precisely, given a fixed sequence \((x_j) \subseteq \mathbb{X}\) (metrizable topological vector space over \(\mathbb{K}=\mathbb{C}\) or \(\mathbb{R}\)), a sequence \(a = (\alpha_j)\) is said to be restricted universal in \(\mathbb{A}\) (vector subspace of \(\mathbb{K}^\mathbb{N}\)) with respect to \(\mu \subseteq \mathbb{N}\) iff, for all \(x \in \mathbb{X}\), there exists \(\lambda = (\lambda_n) \subseteq \mu\) such that NEWLINENEWLINE\[NEWLINE\text{(i)} \:\sum_{j=0}^{\lambda_n} \alpha_j x_j \to x\text{ for }n \to \infty \qquad\text{and}\qquad \text{(ii)} \: \sum_{j=0}^{\lambda_n} \alpha_j e_j \to a\text{ for } n \to \infty. NEWLINE\]NEWLINE The sequence \(a\) is called upper frequently restricted universal if additionally the upper density of \(\lambda\) with respect to \(\mu\) is 1.NEWLINENEWLINENEWLINENEWLINE First result: If the set \({\mathcal U}_{\mathbb{A}}\) of all the restricted universal sequences \(a\) with respect to \(\mathbb{N}\) is non-empty, then the set \(\widetilde{{\mathcal U}^\mu_{\mathbb{A}}}\) of all upper frequently restricted universal sequences is a \(G_\delta\)-dense subset of \(\mathbb{A}\) and contains, except 0, a dense subspace of \(\mathbb{A}\).NEWLINENEWLINE The universality is called unrestricted and the set of universal elements is in this case simply denoted by \({\mathcal U}\) if condition (ii) is not demanded. NEWLINENEWLINENEWLINENEWLINE Second result: NEWLINE\[NEWLINE\begin{multlined} \Bigg\{ a = (\alpha_j) \in {\mathcal U}\; \Bigg|\; \forall x \in \mathbb{X} \;\; \exists \lambda = (\lambda_n) \subseteq \mathbb{N} \text{ with positive lower density}\\ \text{such that } \sum_{j=0}^{\lambda_n} \alpha_j x_j \to x\text{ for } n \to \infty \Bigg\} = \emptyset.\end{multlined}NEWLINE\]NEWLINE NEWLINENEWLINEFinally the author calls a sequence \(a = (\alpha_j) \in {\mathcal U}\) to be lower frequently universal iff, for all \(x \in \mathbb{X}\) and all \(\varepsilon > 0\), the lower density of the set NEWLINENEWLINENEWLINE\[NEWLINE\Bigg\{n\in\mathbb{N}\; \Bigg|\; \varrho\Big( \sum_{j=0}^n \alpha_j x_j , x\Big) < \varepsilon \Bigg\}\Bigg.NEWLINE\]NEWLINENEWLINENEWLINEis positive. If \({\mathcal U}_{\mathbb{A}} \not= \emptyset\), then the class of these sequences in \(\mathbb{A}\) is of first category in \(\mathbb{A}\).