Product sets in the plane, sets of the form \(A+B\) on the real line and Hausdorff measures (Q1333064)

The author gives negative answers to questions due to Laczkovich and Petruska. Every plane set of positive two-dimensional Lebesgue measure contains a cross product set \(A \times B\) such that \(A\) has positive one- dimensional measure and \(B\) is nonempty and perfect. Laczkovich asked whether \(B\) can be of positive Hausdorff dimension. The author shows that for any Hausdorff measure \(\mu^ h\), \(h\) a right continuous Hausdorff function, there exists a set \(E \subset I^ 2\) of full measure such that if \(A \times B \subset E\), \(\lambda_ 1 (A) > 0\), and the sets \(A\), \(B\) are measurable then \(B\) has zero \(\mu^ h\)-measure. The question of Petruska concerns line sets \(A,B\). If \(\lambda_ 1 (B) > 0\) and \(A \cap I\) has Hausdorff dimension \(d\) for any interval \(I\), is it true that the Hausdorff dimension of the complement \((A + B)^ c\) cannot be bigger than \(1-d\)? The author gives an example of a set \(B\) with positive Lebesgue measure, and a set \(A\) satisfying the above conditions with \(d = 1\) but the Hausdorff dimension of \((A + B)^ c\) is 1.