Directional discrepancy in two dimensions

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy

On the minimum degree of minimally t t -tough, claw-free graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and deletion of any edge from $G$ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $t$-tough graph is $\lceil 2t\rceil$ and proved that the minimum degree of minimally $\frac{1}2$-tough and $1$-tough, claw-free graphs is 1 and 2, respectively. We have show that every minimally $3/2$-tough, claw-free graph has a vertex of degree of $3$. In this paper, we give an upper bound on the minimum degree of minimally $t$-tough, claw-free graphs for $t\geq 2$