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 distribution. When ∂ is 2 regardless curvature, that for every set P N points in T have the sharp bound ∫ 0 1 | card ( ∩ τ + t ) − d ⩾ c / . only piecewise not polygon 5 also give whole range intermediate results between Our proofs depend lemma Cassels-Montgomery, ad hoc constructions finite point sets, geometric type estimate average decay Fourier transform characteristic function