A homogeneous method in summation with log factors and its application

TL;DR

This paper presents a homogeneous method for summations with log factors and applies it to analyze the asymptotics of distance energy in square lattices, contributing to the understanding of Erdős's distinct distances conjecture.

Contribution

It introduces a new homogeneous summation method with log factors and applies it to a geometric problem related to Erdős's conjecture.

Findings

Derived asymptotic main terms for distance energy in square lattices

Connected summation techniques with geometric distance problems

Provided insights towards Erdős's distinct distances conjecture

Abstract

We introduce a homogeneous method to deal with summations with homogeneous factors. Then we use it to compute main terms in the asymptotics of distance energy of square lattices in circles, which relates to the conjecture of distinct distances by Erdos.