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Sparse juntas on the biased hypercube

Irit Dinur, Yuval Filmus, Prahladh Harsha

TL;DR

This work extends foundational results on low-degree Boolean functions to the $p$-biased hypercube by proving a structure theorem: any Boolean function that is $\varepsilon$-close to degree $d$ in $L_2$ is close to a sparse junta, with explicit dependence on the bias $p$ and the degree. The authors introduce an explicit family $\mathcal{F}_{d,\varepsilon,p}$ of degree-$d$ sparse juntas and prove a corresponding monotone version linking to sparse DNFs, both achieving optimal or near-optimal dependencies via minimal non-Boolean inputs and a reverse union bound. The proof strategy reduces the biased problem to unbiased instances, applies the Kindler–Safra theorem on each unbiased subspace, and then aggregates via a junta agreement theorem to yield a global sparse junta; the $L_2$ closeness follows from an $L_0$ control through a reverse union bound. These results yield robust, bias-tolerant structure and approximation theorems with potential implications for threshold phenomena and related combinatorial domains. Overall, the paper provides sharp, constructive characterizations of functions near low-degree in the biased setting and tight links between degree, sparsity, and Boolean approximation.

Abstract

We give a structure theorem for Boolean functions on the $p$-biased hypercube which are $ε$-close to degree $d$ in $L_2$, showing that they are close to sparse juntas. Our structure theorem implies that such functions are $O(ε^{C_d} + p)$-close to constant functions. We pinpoint the exact value of the constant $C_d$. We also give an analogous result for monotone Boolean functions on the biased hypercube which are $ε$-close to degree $d$ in $L_2$, showing that they are close to sparse DNFs. Our structure theorems are optimal in the following sense: for every $d,ε,p$, we identify a class $\mathcal{F}_{d,ε,p}$ of degree $d$ sparse juntas which are $O(ε)$-close to Boolean (in the monotone case, width $d$ sparse DNFs) such that a Boolean function on the $p$-biased hypercube is $O(ε)$-close to degree $d$ in $L_2$ iff it is $O(ε)$-close to a function in $\mathcal{F}_{d,ε,p}$.

Sparse juntas on the biased hypercube

TL;DR

This work extends foundational results on low-degree Boolean functions to the -biased hypercube by proving a structure theorem: any Boolean function that is -close to degree in is close to a sparse junta, with explicit dependence on the bias and the degree. The authors introduce an explicit family of degree- sparse juntas and prove a corresponding monotone version linking to sparse DNFs, both achieving optimal or near-optimal dependencies via minimal non-Boolean inputs and a reverse union bound. The proof strategy reduces the biased problem to unbiased instances, applies the Kindler–Safra theorem on each unbiased subspace, and then aggregates via a junta agreement theorem to yield a global sparse junta; the closeness follows from an control through a reverse union bound. These results yield robust, bias-tolerant structure and approximation theorems with potential implications for threshold phenomena and related combinatorial domains. Overall, the paper provides sharp, constructive characterizations of functions near low-degree in the biased setting and tight links between degree, sparsity, and Boolean approximation.

Abstract

We give a structure theorem for Boolean functions on the -biased hypercube which are -close to degree in , showing that they are close to sparse juntas. Our structure theorem implies that such functions are -close to constant functions. We pinpoint the exact value of the constant . We also give an analogous result for monotone Boolean functions on the biased hypercube which are -close to degree in , showing that they are close to sparse DNFs. Our structure theorems are optimal in the following sense: for every , we identify a class of degree sparse juntas which are -close to Boolean (in the monotone case, width sparse DNFs) such that a Boolean function on the -biased hypercube is -close to degree in iff it is -close to a function in .

Paper Structure

This paper contains 46 sections, 34 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof sketch
  4. From $L_0$ to $L_2$ using the reverse union bound
  5. Preliminaries
  6. Monotone functions
  7. Biased measure
  8. Closeness
  9. Monomial expansion
  10. Low degree functions
  11. Function restriction
  12. Function substitution
  13. Boolean functions
  14. Unbiased structure theorems
  15. Granularity
  16. ...and 31 more sections

Key Result

Theorem 1.1

Suppose that $f\colon (\{0,1\}^n, \mu_p) \to \{0,1\}$ is $\varepsilon$-close to degree $d$, where $p \leq 1/2$. Then $f$ is $O(\varepsilon)$-close to $\mathop{\mathrm{round}}\nolimits(g, \{0,1\})$, where $g$ is a degree $d$ polynomial satisfying the following properties, for some constant $C$ depend Conversely, if $g$ is a degree $d$ polynomial satisfying these properties then $\mathop{\mathrm{rou

Theorems & Definitions (40)

  • Theorem 1.1: Main
  • Theorem 1.2
  • Theorem 1.3: Junta approximation
  • Theorem 1.4
  • Theorem 1.5: Kindler--Safra
  • Theorem 1.6: Junta agreement theorem
  • Lemma 2.1
  • Theorem 2.2: Nisan--Szegedy
  • Theorem 2.3: Kindler--Safra
  • Theorem 2.4
  • ...and 30 more