Delone sets associated with badly approximable triangles

TL;DR

This paper constructs new Delone sets linked to badly approximable numbers, aiming for rotationally invariant diffraction, and analyzes the optimality of tile orientations through solutions to a specific linear equation.

Contribution

It introduces a novel method to associate Delone sets with badly approximable numbers and investigates their geometric properties related to diffraction.

Findings

Identifies exactly two solutions with minimal partial quotients for the linear equation.

Shows the constructed Delone sets are expected to have rotationally invariant diffraction.

Optimizes tile orientation discrepancy using number-theoretic analysis.

Abstract

We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation $x+y+z=1$ where $\pi x$, $\pi y$, $\pi z$ are three angles of a triangle used in the construction and $x$, $y$, $z$ are badly approximable. In particular, we show that there are exactly two solutions that have the smallest partial quotients by lexicographical ordering.