Small Shadow Partitions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913567081496576 |
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| author | Kopparty, Swastik Sha, Harry |
| author_facet | Kopparty, Swastik Sha, Harry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Shadow Partitions Kopparty, Swastik Sha, Harry Computational Complexity Combinatorics F.0; G.2.1 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. |
| title | Small Shadow Partitions |
| topic | Computational Complexity Combinatorics F.0; G.2.1 |
| url | https://arxiv.org/abs/2410.22040 |