A one-dimensional planar Besicovitch-type set

A $\mathrm{\Gamma }$-Besicovitch set is a set which contains a rotated copy of $\mathrm{\Gamma }$ in every direction. Our main result is the construction of a non-trivial $1$-rectifiable set $\mathrm{\Gamma }$ in the plane, for which there exists a 1-dimensional $\mathrm{\Gamma }$-Besicovitch set.