Paper detail

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

In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos&#39;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&#39;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 $\{(i,j) \mid 0\le i\le n^{1-α},\ 0\le j\le n^α\}$, for some $0<α<1/2$, and show that the number of distinct distances in such a lattice is $Θ(n)$. In a sense, our proof &#34;bypasses&#34; 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.