In a MO question here @IosifPinelis shows that the ratio of expectations $ mathbb {E} (A) / mathbb {E} (B) $ the largest (speak $ A $) and smallest (speak $ B $) Gap resulting from $ n $ uniform random variables $ (0.1) $ tends to infinity as $ n rightarrow infty. $

He has also shown this in an earlier question related to the above question $ mathbb {E} (A / B) $ goes to infinity as $ n rightarrow infty. $

I have a related question:

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

*What is the fastest growing sequence? $ ell (n) $ so that
$$ lim_ {n rightarrow infty} frac {G _ { ell (n)}} {G _ {(1)}} < infty $$*