Polynomial‐order oscillations in geometric discrepancy

Abstract Let be a convex body, and for a positive integer , let be a configuration of points in . The discrepancy of with respect to is defined by and one may estimate how deviates from uniformity by averaging the latter quantity over a family of sets. When considering quadratic averages over translated and dilated copies of , one gets the homothetic quadratic discrepancy We investigate the behaviour of the optimal homothetic quadratic discrepancy , that is Beck and Chen showed that the optimal h.q.d . of convex polygons has an order of growth of , and more recently, Brandolini and Travaglini proved that the optimal h.q.d . of planar convex bodies with a boundary has an order of growth of . We show that, in general, a single order of growth for the optimal h.q.d . need not exist. First, by an implicit geometric construction of , we obtain prescribed oscillations between and . Second, by a subtler design of and via Fourier‐analytic methods, we obtain prescribed polynomial‐order oscillations in the range with . Moreover, we show that the set of planar convex bodies whose optimal h.q.d . does not admit a single order of growth is residual in the (Hausdorff) metric space of planar convex bodies.