Fix integers 1 < k < n. For V ∈ G(k, n), let PV: ℝn → V be the orthogonal projection. For V ∈ G(k, n), define the map<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11854_2025_363_Article_Equa.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="205" /> </MediaObject> <EquationSource Format="TEX">$\matrix{{{\pi _V}:A\left({1,n} \right) \to A\left({1,V} \right)\bigsqcup V} \cr \;\;\;\;\;\;\;\;\;{\ell \mapsto {P_V}\left(\ell \right).}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mtable> <mtr> <mtd> <mrow> <mrow> <msub> <mi>π</mi> <mi>V</mi> </msub> </mrow> <mo>:</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mo>⨆</mo> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow> <mi>ℓ</mi> <mo stretchy="false">↦</mo> <mrow> <msub> <mi>P</mi> <mi>V</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>ℓ</mi> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </math></EquationSource> </Equation>For any 0 < a < dim(A(1, n)), we find the optimal number s(a) such that the following is true. For any Borel set A ⊂ A(1, n) with dim(A) = a, we have<Equation ID="Equb"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11854_2025_363_Article_Equb.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="306" /> </MediaObject> <EquationSource Format="TEX">$\text{dim}(\pi_{V}(\mathbf{A}))\geq s(a),\;\;\text{for}\;\text{a.e.}\;\;V\in G(k,n).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mtext>dim</mtext> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mtext>for</mtext> <mspace width="thickmathspace" /> <mtext>a.e.</mtext> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>V</mi> <mo>∈</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </math></EquationSource> </Equation>When A(1, n) is replaced by A(0, n) = ℝn, it is the classical Marstrand projection theorem, for which s(a) = min{k, a}. A new ingredient of the paper is the Fourier transform on the affine Grassmannian.

A Marstrand projection theorem for lines

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发表日期：

2025年1月21日

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Shengwen Gan

摘要

Fix integers 1 < k < n. For V ∈ G(k, n), let P_V: ℝⁿ → V be the orthogonal projection. For V ∈ G(k, n), define the map $\matrix{{{\pi _V}:A\left({1,n} \right) \to A\left({1,V} \right)\bigsqcup V} \cr \;\;\;\;\;\;\;\;\;{\ell \mapsto {P_V}\left(\ell \right).}}$ $\begin{matrix} π_{V} : A (1, n) \to A (1, V) ⨆ V \\ ℓ \mapsto P_{V} (ℓ) . \end{matrix}$

For any 0 < a < dim(A(1, n)), we find the optimal number s(a) such that the following is true. For any Borel set A ⊂ A(1, n) with dim(A) = a, we have $\text{dim}(\pi_{V}(\mathbf{A}))\geq s(a),\;\;\text{for}\;\text{a.e.}\;\;V\in G(k,n).$ $dim (π_{V} (A)) \geq s (a), for a.e. V \in G (k, n) .$

When A(1, n) is replaced by A(0, n) = ℝⁿ, it is the classical Marstrand projection theorem, for which s(a) = min{k, a}. A new ingredient of the paper is the Fourier transform on the affine Grassmannian.