For $0<\delta\ll 1$ we define a $\delta$-tube to be any rectangular box $T$ in $\mathbb{R}^d$ with $d-1$ sides length $\delta$ and one side of length $1$, observe that such tubes have volume $\delta^{d-1}$. Let $\mathbb{T}$ be an arbitrary collection of such $\delta$-tubes whose orientations form a $\delta-$ separated set of points on $\mathrm{S}^{d-1}$(i.e. for every direction $e_j, e_k, |e_j-e_k|\leq \delta$). If we use $\#\mathbb{T}$ to denote the cardinality of $\mathbb{T}$ , and $\chi_{T}$ to denote the indicator function of $T$, then the linear Kakeya conjecture says that for each $\frac{d}{d-1}<q\leq\infty$, there is a constant, independent of $\delta$ and the collection $\mathbb{T}$, such that $$ \|\sum_{T\in\mathbb{T}}\chi_{T}\|_{L^{q}(\mathbb{R}^d)}\leq C\delta^{(d-1)/q}(\#\mathbb{T})^{1/q} $$ My first question is that which example can show that the range $\frac{d}{d-1}<q\leq\infty$ is necessary?
Another question is the case $q=\infty$, then this estimate suggest that the LHS should be bounded, which is confusing me a little. Consider a collection of tubes passing through the origin and let the origin be the center of all tubes, then since the intersection of all tubes contains a ball of radius $\delta$, and the number of such tubes can be say $\delta^{-(d-1)}$, then we have $\|\sum_{T\in\mathbb{T}}\chi_{T}\|_{L^{\infty}(\mathbb{R}^d)}\approx\delta^{-(d-1)}$, which is not uniformly bounded, so where am I wrong?