We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When ∂C is C2 regardless of curvature, we prove that for every set PN of N points in T2 we have the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN1/2. When ∂C is only piecewise C2 and is not a polygon we prove the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN2/5. We also give a whole range of intermediate sharp results between N2/5 and N1/2. 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.