Separated sets and the Falconer conjecture for polygonal norms

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms in which the unit ball is a polygon with 2K sides. We prove that for any such norm there is a set of Hausdorff dimension K/(K-1) whose distance set has Lebesgue measure 0.