A note on the unit distance problem for planar configurations with Q-independent direction set

TL;DR

This paper determines the maximum number of unit distances among n points in the plane with Q-independent directions, providing explicit formulas and constructions, and linking the problem to combinatorial and geometric structures.

Contribution

It establishes an explicit formula for T(n), the maximum unit distances with Q-independent directions, and constructs point sets achieving this maximum, connecting to lattice and hypercube graph properties.

Findings

T(n+1)-T(n) equals the Hamming weight of n

T(n) is asymptotically Θ(n log n)

Constructed point sets achieve the maximum T(n)

Abstract

Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\mathbb{Q}$-independent. This is related to the Erdos unit distance problem but with a simplifying additional assumption on the direction set which holds "generically". We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e., the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In particular $T(n)$ is $\Theta(n log(n))$. Furthermore we describe a process to construct a set of $n$ points in the plane with $\mathbb{Q}$-independent unit length direction set which achieves exactly $T(n)$ unit distances. In the process of doing this, we show $T(n)$ is also the same as the maximum number of edges a subset of vertices of size $n$ determines in either the countably infinite lattice $\mathbb{Z}^{\infty}$ or the infinite hypercube graph $\{0,1\}^{\infty}$. The problem of determining T(n) can be viewed as either a type of packing or isoperimetric problem.