<p>In this work, we prove that the complement of the Brjuno set in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> has zero <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation>-capacity with respect to the kernel <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k_\sigma (z,\xi )=\Vert z-\xi \Vert ^{-2n+2}|\log {\Vert z-\xi \Vert |^{\sigma }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mi>σ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>z</mi> <mo>-</mo> <mi>ξ</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mo>-</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">|</mo> <mo>log</mo> </mrow> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>z</mi> <mo>-</mo> <mi>ξ</mi> <mo stretchy="false">‖</mo> <mo stretchy="false">|</mo> </mrow> <mi>σ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma >n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, it follows that it has zero <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation>-Hausdorff measure with respect to the gauge function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$h_\delta (t)=t^{2n-2}|\log {t}|^{-\delta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>δ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>t</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mo>log</mo> <mi>t</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>δ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, for any <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta >n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This generalizes a previous result of A. Sadullaev and the second author in dimension one to higher dimensions.</p>

Capacity Dimension of the Brjuno Set in ${\mathbb {C}}^n$

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发表日期：

2026年6月15日

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Nurali Akramov,
Karim Rakhimov

摘要

In this work, we prove that the complement of the Brjuno set in $\mathbb {C}^n$ $C^{n}$ has zero $C_\sigma $ $C_{σ}$ -capacity with respect to the kernel $k_\sigma (z,\xi )=\Vert z-\xi \Vert ^{-2n+2}|\log {\Vert z-\xi \Vert |^{\sigma }}$ $k_{σ} (z, ξ) = {‖ z - ξ ‖}^{- 2 n + 2} | log {‖ z - ξ ‖ |}^{σ}$ for any $\sigma >n$ $σ > n$ . In particular, it follows that it has zero $h_\delta $ $h_{δ}$ -Hausdorff measure with respect to the gauge function $h_\delta (t)=t^{2n-2}|\log {t}|^{-\delta }$ $h_{δ} (t) = t^{2 n - 2} {| log t |}^{- δ}$ , for any $\delta >n+1$ $δ > n + 1$ . This generalizes a previous result of A. Sadullaev and the second author in dimension one to higher dimensions.

Capacity Dimension of the Brjuno Set in \({\mathbb {C}}^n\)

摘要