A New Lower Bound for the Distinct Distance Constant

TL;DR

This paper introduces a new $B_2$-sequence with a larger reciprocal sum and refines estimates for the reciprocal sums of known sequences, advancing understanding of the Distinct Distance Constant.

Contribution

It presents a novel $B_2$-sequence with a higher reciprocal sum and offers improved estimations for existing sequences' reciprocal sums, contributing to lower bounds for the Distinct Distance Constant.

Findings

A new $B_2$-sequence with larger reciprocal sum

More precise estimates for Mian-Chowla and Zhang sequences

Clarification that the reciprocal sum of Zhang sequence differs from the Distinct Distance Constant

Abstract

The reciprocal sum of Zhang sequence is not equal to the Distinct Distance Constant. This note introduces a $B_2$-sequence with larger reciprocal sum, and provides a more precise estimation of the reciprocal sums of Mian-Chowla sequence and Zhang sequence.