In an MO question [here](https://mathoverflow.net/questions/339164/on-the-largest-and-smallest-spacings-for-the-uniform-distribution) @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables on $(0,1]$ tends to infinity as $n\rightarrow \infty.$


In an earlier question, linked in the above question, he also showed that $\mathbb{E}(A/B)$ equals to infinity for all $n>2.$


I have a related question (thanks @YuvalPeres for your comments): 


*Let $G\_{(1)}$ be the smallest, $G\_{(2)}$ be the second smallest, etc. and let $G\_{(n)}$ be the largest gap.* 


*What is the fastest growing sequence $\ell(n)$ such that 
$$\lim\_{n\rightarrow \infty} \frac{\mathbb{E} G\_{(\ell(n))}}{\mathbb{E} G\_{(1)}}<\infty?$$*


**Edit:** Simulations seem to confirm the proof by @MattF. that $\ell(n)=2$ yields $7/3.$ 



On the convergence of the ratio of order statistics of gaps induced by $n$ uniform points on $[0,1].$
It is [now shown](https://works.bepress.com/iosif-pinelis/17/) that for $i=1,\dots,n-1$
\begin{equation}\label{eq:EG}
 E G\_{n-1:i}=\frac{H\_{n-1}-H\_{n-1-i}}{n+1}, 
\end{equation}
where $G\_{n-1:i}$ is the $i$th smallest value among the gaps $G\_1,\dots,G\_{n-1}$ defined in the [linked post](https://mathoverflow.net/questions/339164/on-the-largest-and-smallest-spacings-for-the-uniform-distribution) and 
\begin{equation\*}
 H\_k:=1+\frac12+\dots+\frac1k 
\end{equation\*}
is the $k$th harmonic number, with $H\_0:=0$. 


In particular, this confirms my conjecture made in an above comment that 
$$r\_{n-1:i}:=\frac{E G\_{n-1:i}}{E G\_{n-1:1}}\sim i
$$
if $1\le i=o(n)$. 


It also follows that $r\_{4-1:2}=5/2$ and $r\_{5-1:2}=7/3$. This seems to differ a bit from Matt's results, perhaps because Matt used notations differing from those in the [linked post](https://mathoverflow.net/questions/339164/on-the-largest-and-smallest-spacings-for-the-uniform-distribution). 


