Bounds for the Regularity Radius of Delone Sets

Abstract

Abstract Delone sets are discrete point sets X in $${\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:math> characterized by parameters ( r , R ), where (usually) 2 r is the smallest inter-point distance of X , and R is the radius of a largest “empty ball” that can be inserted into the interstices of X . The regularity radius $${\hat{\rho }}_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>d</mml:mi></mml:msub></mml:math> is defined as the smallest positive number $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> such that each Delone set with congruent clusters of radius $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that $${\hat{\rho }}_{d}={\textrm{O}(d^2\log _2 d)}R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mover><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>d</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mtext>O</mml:mtext><mml:mo>(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mo>log</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math> as $$d\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , independent of r . This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2 r and those with full-dimensional sets of d -reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that $${\hat{\rho }}_{d}={\textrm{O}(d\log _2 d)}R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mover><mml:mi>ρ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>d</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mtext>O</mml:mtext><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mo>log</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math> as $$d\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , independent of r .

Locations

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