Intersecting Lines Modulo One

A straight line in the plane of gradient α reduces modulo the integer lattice to a set of parallel lines across the unit square. When α is irrational, these lines are dense in the unit square. Two straight lines with different irrational gradients α and β give two dense families of parallel lines, whose intersections are uniformly distributed in the unit square. The discrepancy bounds involve continued fractions, or linear forms in α and β.