Asymptotic estimate on the distance energy of lattices

TL;DR

This work probes whether higher moments of the distance-energy in the Elekes-Sharir framework can sharpen Erdős-type bounds on distinct distances. It shows that for the planar square grid, higher energies satisfy $E_k(P) \asymp N^{k+1}(\log N)^{2^{k-1}-1}$, introducing a logarithmic penalty that defeats the hoped improvement over the second moment; in contrast, in $\mathbb{R}^m$ with $m\ge3$, the second moment remains essentially optimal with $E_2(P)\asymp N^{2+(2m-2)/m}$, yielding $d(P)\asymp N^{2/m}$. The paper then extends the analysis to general lattices via arithmetic lattices and Epstein zeta functions, establishing asymptotics for $E_{D,2}(N)$ in the imaginary-quadratic cases and formulating questions about higher-moment zeta functions $Z_{Q,k}(s)$ and their potential functional equations. Overall, the results indicate higher moments do not improve planar Erdős-type bounds and support the optimality of second-moment estimates in higher dimensions, while linking lattice distance problems to classical number theory and Epstein zeta theory.

Abstract

Since the well-known breakthrough of L. Guth and N. Katz on the Erdos distinct distances problem in the plane, mainstream of interest is aroused by their method and the Elekes-Sharir framework. In short words, they study the second moment in the framework. One may wonder if higher moments would be more efficient. In this paper, we show that any higher moment fails the expectation. In addition, we show that the second moment gives optimal estimate in higher dimensions.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Proposition 3.1
• Lemma 3.2: (7.20) of Wilson
• Lemma 3.3: Theorem 1 of Chapter V in Karatsuba Karatsuba
• proof : Proof of Proposition \ref{['prop-sq sum est.']}
• Theorem 3.4
• Lemma 4.1: Theorem 6.1 of Müller Muller
• Corollary 4.2
• Corollary 4.3