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<p>We extend the notion of hyperuniformity to the projective spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{R}\mathbb{P}^{d-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{C}\mathbb{P}^{d-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{H}\mathbb{P}^{d-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">H</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{O}\mathbb{P}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">O</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We show that hyperuniformity implies uniform distribution and present examples of deterministic point sets as well as point processes which exhibit hyperuniform behaviour.</p>

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Hyperuniform Point Sets on Projective Spaces

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  • 发表日期：

    2026年5月29日

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  • Johann S. Brauchart,
  • Peter J. Grabner

摘要

We extend the notion of hyperuniformity to the projective spaces \(\mathbb{R}\mathbb{P}^{d-1}\) R P d - 1 , \(\mathbb{C}\mathbb{P}^{d-1}\) C P d - 1 , \(\mathbb{H}\mathbb{P}^{d-1}\) H P d - 1 , and \(\mathbb{O}\mathbb{P}^2\) O P 2 . We show that hyperuniformity implies uniform distribution and present examples of deterministic point sets as well as point processes which exhibit hyperuniform behaviour.