A claimed construction of a 1-rectifiable set Γ with a 1-dimensional Γ-Besicovitch set fails because the defining sequences cannot exist.
Mattila, Geometry of sets and measures in Euclidean spaces
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A one-dimensional planar Besicovitch-type set
A claimed construction of a 1-rectifiable set Γ with a 1-dimensional Γ-Besicovitch set fails because the defining sequences cannot exist.