Graph-null sets

Suppose a subset E of the plane can be moved, using vertical and horizontal translations, such that its final position is E+(1,0), and the area touched by the points of E during the horizontal translations is arbitrarily small. Then we say that E has the translational Kakeya property (K^t). Although at first sight the property (K^t) sounds rather paradoxical, as it turns out, an even stronger property called the graph-null property is shared by many curves, including most absolutely continuous curves. I present some unpublished results on graph null sets obtained in 2015 by Viktor Kiss, András Máthé and myself, and mention several open problems.