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 upper Banach (or uniform) density on $\mathbf N^+$, viz. the function $$\mathcal P(\mathbf N^+) \to \mathbf R: X \mapsto \lim_{n \to \infty} \frac{\max_{m \ge 1} |X \cap [m+1,m+n]|}{n},$$ and $\pi_X$ denotes, for each $X \subseteq \mathbf N^+$, the function $\mathbf N^+ \to \mathbf N: n \mapsto |X \cap [1,n]|$ (that is, the counting function of $X$), while $\log^{[k]}$ means, for each $k \in \mathbf N$, the $k$-th iterate of the natural logarithm.
I don't have a precise motivation for this, I'd just like to understand what is known about questions along this line of thought.
For what it is worth, let me note that $k = 1$ can't work, since the set of primes has zero density uniformly with respect to the actual choice of a subadditive and translational invariant function $\mu^\ast: \mathcal P(\mathbf N^+) \to \mathbf R$ such that $\mu^\ast(\mathbf N^+) = 1$ and $\mu^\ast(q \cdot X) = \frac{1}{q} \mu^\ast(X)$ for all $X \subseteq \mathbf N^+$ and $q \in \mathbf N^+$ (the upper Banach density on $\mathbf N^+$ belongs to this class).
