Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

• We present sequences in the plane with low discrepancy.
• The discrepancy is with respect to a smooth convex set intersected with rectangles.
• This allows to numerically approximate integrals of piecewise smooth functions.
• Thanks to a general Erdős–Turán inequality we avoid using isotropic discrepancy.
• The construction of the sequence is based on simultaneous diophantine approximation.

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R2R2. The proof depends on simultaneous diophantine approximation and a general version of the Erdős–Turán inequality.