Erdős distinct distances problem

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 and almost proven by Larry Guth and Nets Katz in 2015.

The conjecture
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
 * $$\sqrt{n-3/4}-1/2\leq g(n)\leq c n/\sqrt{\log n}$$

for some constant $$c$$. The lower bound was given by an easy argument. The upper bound is given by a $$\sqrt{n}\times\sqrt{n}$$ square grid. For such a grid, there are $$O( n/\sqrt{\log n})$$ 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) $$g(n) = \Omega(n^c)$$ holds for every $c < 1$.

Partial results
Paul Erdős' 1946 lower bound of $g(n) = &Omega;(n^{1/2})$ was successively improved to:
 * $g(n) = &Omega;(n^{2/3})$ by Leo Moser in 1952,
 * $g(n) = &Omega;(n^{5/7})$ by Fan Chung in 1984,


 * $g(n) = &Omega;(n^{4/5}/log n)$ by Fan Chung, Endre Szemerédi, and William T. Trotter in 1992,
 * $g(n) = &Omega;(n^{4/5})$ by László A. Székely in 1993,
 * $g(n) = &Omega;(n^{6/7})$ by József Solymosi and Csaba D. Tóth in 2001,
 * $g(n) = &Omega;(n^{(4e/(5e &minus; 1)) &minus; ɛ})$ by Gábor Tardos in 2003,
 * $g(n) = &Omega;(n^{((48 &minus; 14e)/(55 &minus; 16e)) &minus; ɛ})$ by Nets Katz and Gábor Tardos in 2004,
 * $g(n) = &Omega;(n/log n)$ by Larry Guth and Nets Katz in 2015.

Higher dimensions
Erdős also considered the higher-dimensional variant of the problem: for $$d\ge 3$$ let $$g_d(n)$$ denote the minimal possible number of distinct distances among $$n$$ points in $$d$$-dimensional Euclidean space. He proved that $$g_d(n)=\Omega(n^{1/d})$$ and $$g_d(n)=O(n^{2/d})$$ and conjectured that the upper bound is in fact sharp, i.e., $$g_d(n)=\Theta(n^{2/d})$$. József Solymosi and Van H. Vu obtained the lower bound $$g_d(n)=\Omega(n^{2/d - 2/d(d+2)})$$ in 2008.