Type: Article
Publication Date: 1994-01-01
Citations: 8
DOI: https://doi.org/10.1090/s0002-9947-1994-1162100-2
The Hexagonal Packing Lemma of Rodin and Sullivan [6] states that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{s_n} \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Rodin and Sullivan conjectured that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline equals upper O left-parenthesis 1 slash n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{s_n} = O(1/n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This has been proved by Z-Xu He [2]. Earlier, the present author proved the conjecture under some additional restrictions [1]. In the following we are able to remove these restrictions, and thus give an alternative proof of the RS conjecture. The proof is based on our previous article [1]. It is completely different from the proof of He, and it is mainly based on discrete potential theory, as developed by Rodin for the hexagonal case [4].