A Curve with No Simple Crossings by Segments

We construct a closed Jordan curve γ ⊂ R so that γ∩S is uncountable whenever S is a line segment whose endpoints are contained in different connected components of R \ γ. We say that a Jordan arc σ ⊂ R crosses a compact set K ⊂ R if the two endpoints of σ are in different connected components of R \ K. Clearly any arc crossing K must intersect K in at least one point of K. If the intersection consists of exactly one point, we say K has a simple crossing by σ. In this note we answer a question of Percy Deift by constructing a closed Jordan curve γ ⊂ R so that γ∩S is uncountable whenever S is a line segment crossing γ, i.e., γ has no simple crossings by a line segment. Very likely, such examples are known to a variety of people, but I am not aware of a reference in the literature. We will construct a sequence of closed Jordan curves {γn} and a decreasing sequence of positive real numbers {ǫn} ց 0 so that so that if we set Γn = {z ∈ R 2 : dist(z, γn) ≤ ǫ},