Linearithmic lower bound for 1-D "distinct" closest pair of points problem
Given $n$ integer points $p_1,\ldots,p_n$, not necessarily distinct, consider the set of points $2np_1+1,2np_2+2,\ldots,2np_n+n$. The new points are distinct by construction, and the minimal distance is smaller than $n$ iff the original points were not distinct. So an algorithm for your problem can be used to solve element distinctness.