Small Shadow Partitions

TL;DR

This paper constructs partitions of high-dimensional cubes into many parts with small projections, advancing understanding of geometric partitions and their relation to influences of Boolean functions, answering a key open question.

Contribution

It provides a new partitioning method for the cube with many parts and small projections, extending previous bounds from $O(\sqrt{n})$ to $2^{o(n)}$, and links to a conjecture related to the KKL theorem.

Findings

Partition exists for $c$ up to $2^{o(n)}$ with small projections

Construction relates to influences of Boolean functions

Proposes a conjecture implying the KKL theorem

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.