  We consider a planar convex body $C$ and we prove several analogs of Roth's
theorem on irregularities of distribution. When $\partial C$ is $\mathcal{C}%
^{2}$ regardless of curvature, we prove that for every set $\mathcal{P}_{N}$ of
$N$ points in $\mathbb{T}^{2}$ we have the sharp bound \[
\int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left(
\mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda
^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\geqslant cN^{1/2}\;.
\] When $\partial C$ is only piecewise $\mathcal{C}^{2}$ and is not a polygon
we prove the sharp bound% \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert
\mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right)
\right) -\lambda ^{2}N\left\vert C\right\vert \right\vert
^{2}~dtd\lambda\geqslant cN^{2/5}. \] 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 $C$.
