We complete the proof of the upper bound ρ^3≤10R for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all 10R-clusters of a Delone set X with parameters (r,R) are equivalent, then X is a regular system.
Dolbilin, Nikolay; Garber, Alexey; Leopold, Undine; and Schulte, Egon, "On the regularity radius of Delone sets in R3" (2019). Mathematical and Statistical Sciences Faculty Publications and Presentations. 80.