Strong Parallel Repetition Theorem for Free Projection Games
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Spherical cubes: optimal foams from computational hardness amplification
Communications of the ACM
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What is the least surface area of a shape that tiles $\R^d$ under translations by $\Z^d$? Any such shape must have volume~$1$ and hence surface area at least that of the volume-$1$ ball, namely $\Omega(\sqrt{d})$. Our main result is a construction with surface area $O(\sqrt{d})$, matching the lower bound up to a constant factor of $2\sqrt{2\pi/e} \approx 3$. The best previous tile known was only slightly better than the cube, having surface area on the order of $d$. We generalize this to give a construction that tiles $\R^d$ by translations of any full rank discrete lattice $\Lambda$ with surface area $2 \pi \fnorm{V^{-1}}$, where $V$ is the matrix of basis vectors of $\Lambda$, and $\fnorm{\cdot}$ denotes the Frobenius norm. We show that our bounds are optimal within constant factors for rectangular lattices. Our proof is via a random tessellation process, following recent ideas of Raz~\cite{Raz08} in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in $\R^d$ to rectangular lattice points.