Cyclic interval overlap bound ($\mathbb{P}(S\cap T\ne \varnothing )\le 2k/d$)

For independent uniformly placed cyclic intervals of length $k<d$, an overlap is possible at at most $2k-1$ starting-point offsets, or at all $d$ offsets if that number exceeds $d$. Thus the overlap probability is at most $\min \{1,(2k-1)/d\}$. If $R=|S\cap T|$, then $0\le R\le k$ and $\mathbb{E}(e^{aR/k}-1)\le (2k/d)(e^a-1)$ for $a\ge 0$. The overlap law is not the hypergeometric distribution for unrestricted random subsets.