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• W2017601359 abstract "The “visibility” of a planar set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a point <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined as the normalized size of the radial projection of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the unit circle centered at <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Simon and Solomyak in 2006 proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-neighborhoods of such sets. We also prove lower bounds on the visibility of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-neighborhoods of more general sets, based in part on Bourgain’s discretized sum-product estimates." @default.
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• W2017601359 date "2015-04-23" @default.
• W2017601359 modified "2023-09-26" @default.
• W2017601359 title "Quantitative visibility estimates for unrectifiable sets in the plane" @default.
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• W2017601359 doi "https://doi.org/10.1090/tran/6585" @default.
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