§1. Introduction and notation. In [1] and [2], Besicovitch demonstrated that there exist plane sets of measure zero containing line segments (and indeed entire lines) in all directions in the plane. It is natural to ask about the existence of analogous sets in Euclidean spaces of higher dimensions, and in [3] we defined an (n, k)-Besicovitch set to be a subset A of Rn, of n-dimensional Lebesgue measure zero, such that for each k-dimensional subspace Π of Rn, some translate of Π intersects A in a set of positive k-dimensional measure. (Thus Besicovitch's original constructions were for (2,1)-Besicovitch sets.) Recently, Marstrand [5] has shown (by approximating to sets by unions of cubes) that no (3, 2)-Besicovitch sets exist, and simultaneously the author [3] proved using Fourier transform methods that (n, k)- Bsicovitch sets cannot exist if k > ½n.