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Sets with prescribed lower and upper weighted densities. (English) Zbl 1201.11015

Let \(f:{\mathbb N}\to (0,\infty)\) be a weight function. For a subset \(A\subset {\mathbb N}\) let \[ S_f(A,n)=\sum_{\substack{ a\in A\\ a\leq n}} f(a),\quad S_f(n)=\sum_{a\leq n}f(a). \] Define \[ \underline{d}_f(A)=\liminf_{n\to\infty}\frac{S_f(A,n)}{S_f(n)},\quad \overline{d}_f(A)=\limsup_{n\to\infty}\frac{S_f(A,n)}{S_f(n)} \] the lower an upper \(f\)-densities of \(A\), respectively. In the case when \(\underline{d}_f(A)=\overline{d}_f(A)\) we say that \(A\) possesses \(f\)-density \(d_f(A)\). A weight function \(f\) is called regular if \(d_f(a{\mathbb N}+b)=\frac{1}{a}\) for every natural numbers \(a,b\geq 1\). Let \(f,g\) be regular weight functions. Let \(B\subset{\mathbb N}\) such that \[ \underline{d}_f(B)=0,\quad\overline{d}_f(B)=1,\quad d_g(B)=0. \] Let \(0\leq \alpha\leq \beta\leq \gamma\leq \delta\leq 1\). The authors prove that there exists \(A\subset {\mathbb N}\) such that \[ \underline{d}_f(A)=\alpha,\quad\underline{d}_g(A)=\beta,\quad\overline{d}_g(A)=\gamma,\quad\overline{d}_f(A)=\delta. \] Let the functions \(f,g:{\mathbb N}\to(0,\infty)\) satisfy the assumptions \[ \sum_{n=1}^\infty f(n)=\infty,\quad\lim_{n\to\infty}\frac{f(n)}{S_f(n)}=0,\quad g(n)=\frac{f(n)}{\sum_{i=1}^nf(i)}. \] Then there exists \(B\subset{\mathbb N}\) such that \[ \underline{d}_f(B)=0,\quad\overline{d}_f(B)=1,\quad d_g(B)=0. \]

MSC:

11B05 Density, gaps, topology
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