Asymptotic estimate on the distance energy of lattices

TL;DR

This paper investigates the limitations of using higher moments in the Elekes-Sharir framework for the Erdős distinct distances problem, demonstrating that the second moment provides the optimal estimate in higher dimensions.

Contribution

It proves that higher moments do not improve estimates and confirms the second moment's optimality in higher-dimensional settings.

Findings

Higher moments fail to improve the estimate.

The second moment provides the optimal estimate in higher dimensions.

Higher moments do not enhance the framework's efficiency.

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.