Abstract: The \eps-smooth covering density of a unit co-volume lattice L with respect to a convex body K is defined as the minimum volume of a dilate rK such that each point x\in R^n is rK-covered by (1\pm \eps)\Vol(r K) points in L. For any convex body K in R^n we show that for almost all lattices (with respect to the natural Haar-Siegel
measure on the space of lattices) the \eps-smooth covering density is polynomial in n. We also show similar results for random construction A lattices, provided that the ratio between the covering and packing radii of Z^n with respect to K is at most polynomial in n. Our proofs utilize a recent breakthrough of Dhar and Dvir on the discrete Kakeya problem.
Joint work with Oded Regev and Barak Weiss
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