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arXiv · 1306.5469

Quantitative visibility estimates for unrectifiable sets in the plane

M. Bond·I. Laba·J. Zahl
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Abstract

The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) 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 $\delta$-neighbourhoods of such sets. We also prove lower bounds on the visibility of $\delta$-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates

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M. Bond, I. Laba, J. Zahl. 2013-06-23. Quantitative visibility estimates for unrectifiable sets in the plane. https://doi.org/10.1090/tran/6585

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