More unit distances in arbitrary norms

TL;DR

The paper advances the unit distance problem in arbitrary norms on $\mathbb{R}^d$ by showing a universal lower bound $U_{\|\cdot\|}(n) \ge \big(\tfrac{d}{2}-o(1)\big) n \log_2 n$ for every norm, matching the best known upper bound up to $o(1)$ and extending the $d$-dimensional construction via generalized arithmetic progressions and dimension-theoretic tools. It develops a comprehensive framework in which a 2D warm-up is generalized to all dimensions, using shadow boundaries, carefully chosen point sets, and GAP-based counting to guarantee many unit distances. In addition, for $d\ge 3$ and a comeagre set of norms, the unit distance graph contains $K_{d,m}$ for all $m$, established via a local model and Brouwer degree stability, illustrating robust structural richness in typical norms. Altogether, the work tightens asymptotics for arbitrary norms and reveals strong bipartite-complete structures in typical higher-dimensional norms, contributing a blend of combinatorial, topological, and convex-geometry methods.

Abstract

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\geq 3$ and a typical norm on $\mathbb R^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$.

Figures (4)

• Figure 1: Selection of the points $p_i$ and $q_i$.
• Figure 2: Case (5) of the non-overlapping property of $\mathcal{U}$.
• Figure 3: The construction from \ref{['thm:local-model-stable']} in three dimensions. Recall that $\Sigma_1$ and $\Sigma_2$ are the translates of $\Sigma_0$ which send $(p_1;\rho(p_1))$ resp. $(p_2;\rho(p_2))$ to $(0,0)$. The two parabolas through $(p_1;\rho(p_1))$ and $(p_2;\rho(p_2))$ lie on $\Sigma_0$, hence $\Sigma_1 \cap \Sigma_2$ is the parabola through $(0;0)$. The surface $\Sigma_0$ is "crinkled" near the origin so that this parabola cuts it transversally, creating transversal intersections of $\Sigma_0$, $\Sigma_1$ and $\Sigma_2$.
• Figure 4: Construction of $C$ inside of $B'$.

Theorems & Definitions (24)

• Theorem 1.1: Mat11
• Theorem 1.2: ABS25
• Theorem 1.3: ABS25
• Theorem 1.4
• Theorem 1.5
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• Lemma 4.1
• proof