Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf\_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a density, of if they have atoms, but not in the general case.



How often two iid variables are close?
If $\epsilon \geqslant \tfrac{1}{n}$, then
$$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \sum\_{i=1}^n \mathbb{P}(X, Y \in [\tfrac{i-1}{n}, \tfrac{i}{n}]) = \sum\_{i=1}^n (\mathbb{P}(X \in [\tfrac{i-1}{n}, \tfrac{i}{n}]))^2 . $$
It follows that
$$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \frac{1}{n} \biggl(\sum\_{i=1}^n \mathbb{P}(X \in [\tfrac{i-1}{n}, \tfrac{i}{n}])\biggr)^2 = \frac{1}{n} \, . $$
If we choose $n$ so that additionally $\epsilon < \frac{1}{n-1}$, then we obtain
$$ \mathbb{P}(|X-Y|<\epsilon) > \frac{\epsilon}{\epsilon + 1} \, . $$
This leads to the desired result with $c = 1$.


