Exponential odd-distance sets under the Manhattan metric

TL;DR

This paper constructs large sets of points in n-dimensional space with all pairwise Manhattan distances odd integers, revealing unique behaviors of odd-distance sets under the Manhattan metric compared to Euclidean and maximum metrics.

Contribution

It provides a new construction of exponential-sized odd-distance sets in  space, improving previous bounds and proving optimality under integer or half-integer coordinates.

Findings

Constructed a set of 2^n points with all odd Manhattan distances.

Improved the lower bound for odd-distance sets under the Manhattan metric.

Proved the optimality of the construction with integer or half-integer coordinates.

Abstract

We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction.