A LARGE PLANE SET INTERSECTING LINES IN INFINITELY MANY DIRECTIONS IN AT MOST ONE POINT

We prove that for every at most countable family {f(k)(x)} of real functions on [0, 1) there is a single-valued real function F(x), x is an element of [0, 1), such that the Hausdorff dimension of the graph Gamma of F(x) equals 2, and for every C is an element of R and every k, the intersection of Gamma with the graph of the function f(k)(x)+C consists of at most one point. We also construct a family of functions of cardinality continuum and a function F with similar properties.