I'm reading a recent paper "Finding Correlations in Subquadratic Time, with Applications to Learning Parities and the Closest Pair Problem" by Gregory Valiant on finding approximate closest pairs in $R^d$. Valiant says the current upper bound for this problem is $O(n^2)$.
On Wikipedia however it is said that the 2d divide-and-conquer algorithm can be generalized to solving the problem in $O(n\text{log}n)$ for any number of dimensions. There are no direct sources to this, but in the talk page, a few papers are referenced.
Reading those papers, they all seem to refer to Jon Louis Bentley's thesis from 1980 "Multidimensional Divide-and-Conquer". However as far as I can understand, the closest pair problem is hear described as solved in $O(n\text{log}^dn)$ time.
Hence, I feel like I must have absolutely confused myself on the definitions. Can anyone help me figure out what the correct upper bound for this problem really is?
Update: I should of course define the problem properly: Given a set $S$ of $n$ points from $R^d$, find two points $p, q\in S$, $p\neq q$ which minimise $d(p, q)$, where $d$ is euclidean distance. The model is RAM or similar.
