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 $C$.
Categories: math.MG math.NT