[Paper Review] On lattices, distinct distances, and the Elekes-Sharir framework

Javier Cilleruelo, Micha Sharir+1 more|arXiv (Cornell University)|Jun 2, 2013

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper investigates the limitations of the Elekes-Sharir framework in deriving optimal lower bounds for distinct distances in integer lattices. By analyzing Erdős’s construction of a √n×√n lattice, the authors show that the framework’s reliance on the Cauchy-Schwarz inequality inherently restricts it from achieving the conjectured Θ(n/√log n) bound, even if the true bound holds. The paper establishes that the framework’s upper bound on quadruples of equal distances is tight for the square lattice, implying that a major methodological shift—beyond Cauchy-Schwarz—is required to close the gap.

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 $\{(i,j) \mid 0\le i\le n^{1-α},\ 0\le j\le n^α\}$, for some $0

Motivation & Objective

• To analyze the limitations of the Elekes-Sharir framework in deriving optimal lower bounds for distinct distances in point sets.
• To demonstrate that the framework’s use of the Cauchy-Schwarz inequality creates an inherent barrier to achieving the conjectured Θ(n/√log n) lower bound for the square lattice.
• To show that Guth and Katz’s upper bound on the number of quadruples with equal distances is tight for the square lattice, implying the framework cannot improve beyond Ω(n/log n) without major changes.
• To establish that the number of distinct distances in rectangular lattices of the form {(i,j) : 0≤i≤n^{1−α}, 0≤j≤n^α} for 0<α<1/2 is Θ(n), bypassing a deep number-theoretic conjecture.
• To connect this result to a conjecture by Cilleruelo and Granville on the number of lattice points on short arcs of circles, showing that a positive resolution would imply the same bound.

Proposed method

• Reformulating Erdős’s original lower bound for the square lattice within the Elekes-Sharir framework, focusing on counting quadruples (a,p,b,q) with |ap|=|bq|.
• Applying the Cauchy-Schwarz inequality to derive a lower bound on |Q|, the number of such quadruples, in terms of the number of distinct distances x.
• Establishing that for the √n×√n lattice, |Q|=Θ(n³log n), matching Guth and Katz’s upper bound, thus proving the framework’s bound is tight for this instance.
• Using number-theoretic tools, particularly Ramanujan’s results on sums of two squares, to analyze the number of representations of integers as sums of two squares in restricted ranges.
• Analyzing the number of solutions to a²−b² = c²−d² in intervals Iₗ to bound ∑ₘ binom(d(m),2), which counts pairs of representations of the same number.
• Applying a parametrization via quadruples (s₁,s₂,s₃,s₄) satisfying s₁s₂ = a−c, s₃s₄ = a+c, s₁s₃ = b−d, s₂s₄ = b+d to control the number of such representations.

Experimental results

Research questions

• RQ1 Can the Elekes-Sharir framework, as currently structured, achieve the conjectured Θ(n/√log n) lower bound for distinct distances in the square lattice?
• RQ2 Is the upper bound of Θ(n³log n) on the number of quadruples with equal distances tight for the square lattice, and what does this imply about the framework’s limitations?
• RQ3 What is the number of distinct distances in rectangular lattices of the form {(i,j) : 0≤i≤n^{1−α}, 0≤j≤n^α} for 0<α<1/2?
• RQ4 To what extent does the number-theoretic conjecture by Cilleruelo and Granville on short arcs of circles imply the bound D_α(n)=Θ(n) for rectangular lattices?
• RQ5 Can the bound D_α(n)=Θ(n) be derived without assuming the Cilleruelo–Granville conjecture, and if so, how?

Key findings

• The Elekes-Sharir framework cannot achieve the conjectured Θ(n/√log n) lower bound for distinct distances in the square lattice without replacing the Cauchy-Schwarz inequality with a fundamentally different technique.
• For the √n×√n integer lattice, the number of quadruples (a,p,b,q) with |ap|=|bq| is Θ(n³log n), matching Guth and Katz’s upper bound, proving the framework’s bound is tight for this instance.
• The number of distinct distances in rectangular lattices of the form {(i,j) : 0≤i≤n^{1−α}, 0≤j≤n^α} for 0<α<1/2 is Θ(n), independent of α.
• This result is established using number-theoretic estimates on the number of representations of integers as sums of two squares in restricted ranges, relying on Ramanujan’s results.
• The bound D_α(n)=Θ(n) is shown to be equivalent to a special case of the Cilleruelo–Granville conjecture on lattice points on short arcs of circles, and the paper provides a proof that bypasses this conjecture.
• The analysis shows that the number of solutions to a²−b² = c²−d² in intervals Iₗ is O(n²α log²n), leading to the final bound on the number of distinct distances.