In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances.
It was posed by Paul Erdős in 1946[1][2] and almost proven by Larry Guth and Nets Katz in 2015.
[3][4][5] In what follows let g(n) denote the minimal number of distinct distances between n points in the plane, or equivalently the smallest possible cardinality of their distance set.
In his 1946 paper, Erdős proved the estimates for some constant
The lower bound was given by an easy argument.
The upper bound is given by a
square grid.
For such a grid, there are
log n
numbers below n which are sums of two squares, expressed in big O notation; see Landau–Ramanujan constant.
Erdős conjectured that the upper bound was closer to the true value of g(n), and specifically that (using big Omega notation)
holds for every c < 1.
Paul Erdős' 1946 lower bound of g(n) = Ω(n1/2) was successively improved to: Erdős also considered the higher-dimensional variant of the problem: for
denote the minimal possible number of distinct distances among
-dimensional Euclidean space.
He proved that
and conjectured that the upper bound is in fact sharp, i.e.,
József Solymosi and Van H. Vu obtained the lower bound