This work is motivated by the two classical theorems on inscribing rectangles and squares into large subsets of the plane: Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem, we prove that for every Borel subset of the plane with uncountably many vertical sections that are positive (with respect to measure or category) contains a rectangle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$P\times B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>×</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> where P is perfect and B is Borel and positive. We also present a variant of Eggleston Theorem regarding the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma -$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>ideal <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> generated by closed sets of measure zero. Furthermore, we show that every comeager (resp. conull) subset of the plane contains a rectangle <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$[T]\times H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, where T is a splitting tree containing a Silver tree and H is comeager (resp. conull). Additionally, we establish a joint generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$[T]\times H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> modulo diagonal, where T is a uniformly perfect tree, H is comeager (resp. conull) and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$[T]\subseteq H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo>⊆</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>.

Around the Eggleston Theorem

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2025年11月7日

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Marcin Michalski,
Robert Rałowski,
Szymon Żeberski

摘要

This work is motivated by the two classical theorems on inscribing rectangles and squares into large subsets of the plane: Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem, we prove that for every Borel subset of the plane with uncountably many vertical sections that are positive (with respect to measure or category) contains a rectangle $P\times B$ $P \times B$ where P is perfect and B is Borel and positive. We also present a variant of Eggleston Theorem regarding the $\sigma -$ $σ -$ ideal $\mathcal {E}$ $E$ generated by closed sets of measure zero. Furthermore, we show that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ $[T] \times H$ , where T is a splitting tree containing a Silver tree and H is comeager (resp. conull). Additionally, we establish a joint generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ $[T] \times H$ modulo diagonal, where T is a uniformly perfect tree, H is comeager (resp. conull) and $[T]\subseteq H$ $[T] \subseteq H$ .