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• W1529004018 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>C_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 2> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>C_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be Cantor sets embedded in the real line, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau 1> <mml:semantics> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>tau _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau 2> <mml:semantics> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>tau _{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be their respective thicknesses. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau 1 tau 2 greater-than 1> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>tau _{1}tau _{2}>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then it is well known that the difference set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 minus upper C 2> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}-C_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a disjoint union of closed intervals. B. Williams showed that for some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t element-of i n t left-parenthesis upper C 1 minus upper C 2 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>tin operatorname {int} (C_{1}-C_{2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it may be that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 intersection left-parenthesis upper C 2 plus t right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}cap (C_{2}+t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is as small as a single point. However, the author previously showed that generically, the other extreme is true; <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 intersection left-parenthesis upper C 2 plus t right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}cap (C_{2}+t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a Cantor set for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a generic subset of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 minus upper C 2> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}-C_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau 1 tau 2 greater-than 1> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>τ<!-- τ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>tau _{1}tau _{2}>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 intersection left-parenthesis upper C 2 plus t right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}cap (C_{2}+t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a Cantor set for almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 1 minus upper C 2> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{1}-C_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1529004018 created "2016-06-24" @default.
• W1529004018 creator A5052856778 @default.
• W1529004018 date "1999-09-20" @default.
• W1529004018 modified "2023-09-23" @default.
• W1529004018 title "Random intersections of thick Cantor sets" @default.
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