On lattices, distinct distances, and the Elekes-Sharir framework

Abstract: In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos's lower bound for the square lattice, recast in the setup of the so-called Elekes-Sharir framework \cite{ES11,GK11}, and show that, without a major change, this framework \emph{cannot} lead to Erdos's conjectured lower bound. This shows that the upper bound of Guth and Katz \cite{GK11} for the related 3-dimensional line-intersection problem is tight for this instance. The gap between this bound and the actual bound of Erdos arises from an application of the Cauchy-Schwarz inequality (which is an integral part of the Elekes-Sharir framework). Our analysis relies on two number-theoretic results by Ramanujan. We also consider distinct distances in rectangular lattices of the form ${(i,j) \mid 0\le i\le n^{1-\alpha},\ 0\le j\le n^{\alpha}}$, for some $0<\alpha<1/2$, and show that the number of distinct distances in such a lattice is $\Theta(n)$. In a sense, our proof "bypasses" a deep conjecture in number theory, posed by Cilleruelo and Granville \cite{CG07}. A positive resolution of this conjecture would also have implied our bound.