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arXiv · 1804.05035

On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets

Igor A. Baburin·Mikhail Bouniaev·Nikolay Dolbilin·Nikolay Yu. Erokhovets·Alexey Garber·Sergey V. Krivovichev·Egon Schulte
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Abstract

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hatρ_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{ρ_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hatρ_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.

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Igor A. Baburin, Mikhail Bouniaev, Nikolay Dolbilin, Nikolay Yu. Erokhovets, Alexey Garber, Sergey V. Krivovichev, Egon Schulte. 2018-04-13. On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets. https://doi.org/10.1107/s2053273318012135

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