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<p>We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the <i>n</i> generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the numbers <i>n</i> arising as norms of the Eisenstein integers based upon the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Eisenstein norms. We apply this strategy on the hexagonal torus and prove that the number of defects is at most <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(n^{1/4})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>—strictly fewer than surfaces with boundary—and conjecture (based upon the number-theoretic gap conjecture) that it is in fact <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O(\log ^{3/2+\varepsilon } n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mo>log</mo> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere.</p>

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On the Number of Defects in Optimal Quantizers on Closed Surfaces: The Hexagonal Torus

  • 期刊论文

  • 发表日期：

    2026年1月29日

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  • Jack Edward Tisdell,
  • Rustum Choksi,
  • Xin Yang Lu

摘要

We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the n generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the numbers n arising as norms of the Eisenstein integers based upon the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Eisenstein norms. We apply this strategy on the hexagonal torus and prove that the number of defects is at most \(O(n^{1/4})\) O ( n 1 / 4 ) —strictly fewer than surfaces with boundary—and conjecture (based upon the number-theoretic gap conjecture) that it is in fact \(O(\log ^{3/2+\varepsilon } n)\) O ( log 3 / 2 + ε n ) . Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere.