Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When partial derivative C is C-2 regardless of curvature, we prove that for every set P-N of N points in T-2 we have the sharp boundintegral(1)(0)integral(T2)vertical bar card (P-N boolean AND(tau C + t)) - tau(2) N vertical bar C vertical bar vertical bar(2) dtd tau >= cN(1/2).When partial derivative C is only piecewise C-2 and is not a polygon we prove the sharp bound.integral(1)(0)integral(T2)vertical bar card (P-N boolean AND(tau C + t)) - tau(2) N vertical bar C vertical bar vertical bar(2) dtd tau >= cN(2/5).We also give a whole range of intermediate sharp results between N-2/5 and N-1/2. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.