On the tube-occupancy of sets in $\mathbb{R}^d$

Call a pair $(s,t) \in [0,d] \times [0,d]$ admissible, if there exists a compact set $K \subset \mathbb{R}^{d}$ and a constant $C > 0$ such that $0 < \mathcal{H}^{s}(K) < \infty$, and $$\mathcal{H}_{s}(K \cap T) \leq Cw(T)^{t}$$ for all tubes $T \subset \mathbb{R}^{d}$ of width $w(T)$. The purpose of this paper is to show that all pairs $(s,s)$ with $s < 1$ are admissible. Combined with previous results, this settles a question of A. Carbery.