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Small Shadow Partitions

Swastik Kopparty, Harry Sha

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.

Small Shadow Partitions

Abstract

We study the problem of partitioning the unit cube into parts so that each -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 `-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 , we show that for even as large as , it is possible to partition into parts so that every -dimensional axis-parallel projection has volume at most . Previously, this was shown by [KN23] for up to . 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.

Paper Structure

This paper contains 13 sections, 21 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Other related work
  3. Preliminaries: partitions, projections, influences
  4. Prior work
  5. A general lower bound from classical projection inequalities
  6. Covering with solid hypercubes
  7. n=3,d=2
  8. Partition Sauer-Shelah
  9. Partitions from low influence functions
  10. Projections of partitions and influences
  11. Main Result: Partitions with small projections from low influence functions
  12. c = 2bn
  13. Proof of the d=1 lower bound

Key Result

Theorem 2.3

Let $f: [0,1]^n \to \left\{0,1\right\}$, such that $\Pr_{x \sim [0,1]^n}[f(x) = 1] = p$, then $\mathop{\mathrm{MaxInf}}\nolimits(f) \geq \Omega(p(1-p)\log(n) / n).$

Theorems & Definitions (32)

  • Theorem 2.3: BKKKL bourgainInfluenceVariablesProduct1992
  • Theorem 3.1: Uniform Cover Inequality (bollobasProjectionsBodiesHereditary1995, Theorem 2)
  • Lemma 3.2
  • proof
  • Lemma 3.3: General Lower Bound
  • proof
  • Corollary 3.4
  • Theorem 3.5: mcelieceCoveringToriSquares1973
  • Theorem 3.6
  • Lemma 4.1: Partition Sauer-Shelah
  • ...and 22 more