[Paper Review] Large zero-free subsets of Z/pZ

Jean‐Marc Deshouillers, Gyan Prakash|ArXiv.org|Jan 23, 2009

Limits and Structures in Graph Theory3 citations

TL;DR

This paper determines the maximum size of zero-free subsets in ℤ/pℤ for large primes p, proving that the largest such subsets have size ⌊√(2p)⌋ − δ(p), where δ(p) ∈ {0,1}. Using trigonometric sums and structural analysis, the authors show that any large zero-free set is, up to scaling, a union of a small negative part and a positive part whose elements sum to at most p−1, confirming the Erdős–Heilbronn conjecture asymptotically and refining earlier bounds on the sumset structure of such sets.

ABSTRACT

A finite subset $A$ of an abelian group $G$ is said to be zero-free if the identity element of $G$ cannot be written as a sum of distinct elements from $A$. In this article we study the structure of zero-free subsets of $Z/pZ$ the cardinality of which is close to largest possible. In particular, we determine the cardinality of the largest zero-free subset of $Z/pZ$, when $p$ is a sufficiently large prime.

Motivation & Objective

• To determine the exact maximum cardinality of zero-free subsets in ℤ/pℤ when p is a sufficiently large prime.
• To characterize the structural form of large zero-free subsets in ℤ/pℤ, particularly their distribution and sumset properties.
• To refine and confirm the Erdős–Heilbronn conjecture on the upper bound of zero-free subset sizes in ℤ/pℤ.
• To establish that large zero-free sets are, up to scaling, unions of a small negative set and a positive set summing to at most p−1.
• To improve upon prior bounds on the error terms in sumset estimates for zero-free sets using advanced trigonometric sum techniques.

Proposed method

• Applies trigonometric sum methods developed by Deshouillers and Freiman to analyze incomplete and zero-free subsets in ℤ/pℤ.
• Uses the canonical lift of elements in ℤ/pℤ to integers in (−p/2, p/2] to study sumset behavior and avoid multiples of p.
• Employs the concept of the sumset 𝒜♯ = {∑b∈ℬ b : ∅ ≠ ℬ ⊂ 𝒜} to define zero-freeness as 0 ∉ 𝒜♯.
• Applies results from Szemerédi and Vu on long arithmetic progressions in sumsets to derive structural constraints on zero-free sets.
• Uses partitioning techniques to split scaled zero-free sets into a negligible negative part and a positive part with bounded sum.
• Applies combinatorial lemmas on sumset coverage (e.g., if a set B ⊂ [1,q] has density >7/8, then [q+1,13/8q] ⊂ 2B) to prove interval containment in sumsets.

Experimental results

Research questions

• RQ1 What is the maximum possible size of a zero-free subset in ℤ/pℤ for a large prime p?
• RQ2 How are large zero-free subsets in ℤ/pℤ structured, particularly after scaling by a unit in ℤ/pℤ?
• RQ3 Can the Erdős–Heilbronn conjecture that |𝒜| ≤ √(2p) be confirmed for sufficiently large primes?
• RQ4 What is the optimal error term in the sumset bounds for large zero-free sets, and can it be improved to O(p^{1/2})?
• RQ5 To what extent can the sumset of a zero-free set cover intervals in ℤ/pℤ, and how does this relate to its structure?

Key findings

• The maximum size of a zero-free subset in ℤ/pℤ is the largest integer k such that k(k+1)/2 ≤ p+1, which equals ⌊√(2p)⌋ − δ(p) with δ(p) ∈ {0,1}.
• For any large zero-free set 𝒜, there exists a unit d ∈ (ℤ/pℤ)× such that d·𝒜 is the disjoint union of a set 𝒜′ with at most 1+δ(p) elements in [−2(1+δ(p)),−1]_p and a set 𝒜′′ ⊂ [1,p/2]_p with ∑|a′′| ≤ p−1.
• The sum of absolute values of elements in d·𝒜 is bounded by p + O(p^{3/4} ln p), and the sum of absolute values of negative elements is O(p^{3/4} ln p), with error terms improved to O(p^{1/2}) by Nguyen, Szemerédi, and Vu.
• The set of positive lifts P(0.9√(2p)) has cardinality 0.9√(2p) − O(e(𝒜)), and its sumset P♯ contains the interval [0.9√(2p)+1, ψ(p)^{1/2}p] for a suitable function ψ(p).
• The number of negative lifts N is O(√e(𝒜)), and N is contained in [1, c₀e(𝒜)] for some absolute constant c₀, implying structural sparsity.
• The sumset of the positive part 𝒜′′ satisfies ∑|a′′| ≤ p−1, and the sumset of the negative part 𝒜′ is contained in a small interval, confirming the structural decomposition of large zero-free sets.