Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Kopparty, Swastik, Sha, Harry
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404341547