We construct a compact subset of $\R$ with Hausdorff dimension 1 that intersects each of its non-identical translates in at most one point. Moreover, one can make the set to be linearly independent over the rationals.
"A 1-Dimensional Subset of the Reals that Intersects Each of its Translates in at Most a Single Point." Real Anal. Exchange 24 (2) 843 - 845, 1998/1999.