Shallow packings, semialgebraic set systems, Macbeath regions and polynomial partitioning

International audienceThe packing lemma of Haussler states that given a set system $(X, R)$ with bounded VC dimension, if every pair of sets in $R$ are 'far apart' (i.e., have large symmetric difference), then $R$ cannot contain too many sets. This has turned out to be the technical foundation for many results in geometric discrepancy using the entropy method (see [Mat99] for a detailed background) as well as recent work on set systems with bounded VC dimension [FPS + ar]. Recently it was generalized to the shallow packing lemma [DEG15, Mus16], applying to set systems as a function of their shallow cell complexity. In this paper we present several new results and applications related to packings: 1. an optimal lower bound for shallow packings, thus settling the open question in Ezra (SODA 2016) and Dutta et al. (SoCG 2015). 2. improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry (Annals of Mathematics, 1952). 3. simplifying and generalizing the main technical tool in Fox et al. (J. of the EMS, 2016). Besides using the packing lemma and a combinatorial construction, our proofs combine tools from polynomial partitioning and the probabilistic method