Small Shadow Partitions

Swastik Kopparty; Harry Sha

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arXiv · 2410.22040

Small Shadow Partitions

Swastik Kopparty·Harry Sha

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Abstract

We study the problem of partitioning the unit cube $[0,1]^n$ into $c$ parts so that each $d$-dimensional axis-parallel projection has small volume. This natural combinatorial/geometric question was first studied by Kopparty and Nagargoje [KN23] as a reformulation of the problem of determining the achievable parameters for seedless multimergers -- which extract randomness from `$d$-where' random sources (generalizing somewhere random sources). This question is closely related to influences of variables and is about a partition analogue of Shearer's lemma. Our main result answers a question of [KN23]: for $d = n-1$, we show that for $c$ even as large as $2^{o(n)}$, it is possible to partition $[0,1]^n$ into $c$ parts so that every $n-1$-dimensional axis-parallel projection has volume at most $(1/c) ( 1 + o(1) )$. Previously, this was shown by [KN23] for $c$ up to $O(\sqrt{n})$. The construction of our partition is related to influences of functions, and we present a clean geometric/combinatorial conjecture about this partitioning problem that would imply the KKL theorem on influences of Boolean functions.

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Swastik Kopparty, Harry Sha. 2024-10-29. Small Shadow Partitions. https://arxiv.org/abs/2410.22040

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