We extend the work of An et al. (Int. Math. Res. Not. 24 (2015) 13623–13652), on bounded orbits of diagonalizable flows on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_3(\mathbb {R})/\textrm{SL}_3(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mtext>SL</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_3(\mathbb {C})/\textrm{SL}_3(\mathcal {O}_{\mathbb {K}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mtext>SL</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi mathvariant="double-struck">K</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation> is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski’s linear forms theorem. We then set up an appropriate Schmidt game in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {C}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {C}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> relative to an approximation by imaginary quadratic rationals in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_815_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>.

Hyperplane absolute winning property of bounded orbits under diagonalizable flows on ${\textbf{SL}}_{\textbf {3}}({\pmb {\mathbb {C}}})\varvec{/}{\textbf{SL}}_{\textbf {3}}({\varvec{\mathcal {O}_{\pmb {\mathbb {K}}}}})$

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发表日期：

2025年3月27日

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Gaurav Sawant

摘要

We extend the work of An et al. (Int. Math. Res. Not. 24 (2015) 13623–13652), on bounded orbits of diagonalizable flows on $\textrm{SL}_3(\mathbb {R})/\textrm{SL}_3(\mathbb {Z})$ ${SL}_{3} (R) / {SL}_{3} (Z)$ to $\textrm{SL}_3(\mathbb {C})/\textrm{SL}_3(\mathcal {O}_{\mathbb {K}})$ ${SL}_{3} (C) / {SL}_{3} (O_{K})$ , where $\mathbb {K}$ $K$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski’s linear forms theorem. We then set up an appropriate Schmidt game in $\mathbb {C}^3$ $C^{3}$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in $\mathbb {C}^3$ $C^{3}$ relative to an approximation by imaginary quadratic rationals in $\mathbb {K}$ $K$ .

Hyperplane absolute winning property of bounded orbits under diagonalizable flows on \({\textbf{SL}}_{\textbf {3}}({\pmb {\mathbb {C}}})\varvec{/}{\textbf{SL}}_{\textbf {3}}({\varvec{\mathcal {O}_{\pmb {\mathbb {K}}}}})\)

摘要