All Questions

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...