Quantitative visibility estimates for unrectifiable sets in the plane

The “visibility” of a planar set S S from a point a a is defined as the normalized size of the radial projection of S S from a a to the unit circle centered at a a . 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 δ δ -neighborhoods of such sets. We also prove lower bounds on the visibility of δ δ -neighborhoods of more general sets, based in part on Bourgain’s discretized sum-product estimates.