A \(1\)-dimensional subset of the reals that intersects each of its translates in at most a single point (Q5932458)

Recalling Mattila's construction of compact sets \(A,B\subset \mathbb R^n\) with Hausdorff dimension \(1\) such that \(A\) intersected with an arbitrary (non-identical) translation of \(B\) consists of at most one point [\textit{P. Mattila}, Acta Math. 152, 77-105 (1984; Zbl 0544.28004)], the author finds a compact set \(A\subset \mathbb R^n\) such that the couple \((A,A)\) enjoys the above property.