Consider that $X_1^{(n)},...,X_n^{(n)}$ are iid uniform random variables on $[0,n]$. For $T >0$, let $N_n(T) = \sup_{t \in [0,n]} \# \{ i: |X_i^{(n)} - t| \leq T \}$ be the maximum number of points in a ball of radius $T$. It seems natural to me that if would hold, for any fixed finite $T$ and $k$, $$ \lim_{n \to \infty} P \left( N_n(T) \geq k \right) = 1. $$ I would also expect more general results to hold, assessing roughly that for $n$ independent random points on a domain of volume $n$, then with large probability there are clusters of closely spaced points.
I tried to google some keywords but did not find literature on this topic. I would appreciate if someone could give me references addressing it.
Thank you.