Spectral Bounds for Antipodal Graphs

TL;DR

This paper establishes spectral bounds relating neighbors and antipodes in point sets with bounded diameter, improving previous asymptotic bounds and extending results to higher dimensions.

Contribution

It provides new spectral bounds for antipodal graphs, achieving the conjectured asymptotic within a polylog factor and generalizing to higher dimensions.

Findings

Ratios of neighbors to antipodes are bounded below by ^{1/2 + o(1)} in 2D.

Improves previous bounds from ^{3/4+o(1)} to the conjectured asymptotic.

Extends results to dimensions d  3/2 and 3(d - 1)/4.

Abstract

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $|x_i - x_j| \leq 1$ for all $1 \leq i,j \leq n$. We show that the ratio of the number of ``neighbors'' (ordered pairs of points with distance $\leq \varepsilon$) to the number of ``antipodes'' (ordered pairs of points with distance $\geq 1 - \varepsilon$) is $\gtrsim\varepsilon^{1/2 + o(1)}$, attaining the conjectured correct asymptotic within a polylog factor and improving the $\gtrsim\varepsilon^{3/4+o(1)}$ bound of Steinerberger (2025). In dimensions $d\ge3$ we prove a similar result with exponent $\min\left\{d-3/2,\ 3(d - 1)/4\right\}$.