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Zero subsums in vector spaces over finite fields

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Pith's one-line read For every fixed dimension d and every epsilon>0, the Olson constant of F_p^d is at most (d-1+epsilon)p for all sufficiently large primes p, settling the Nguyen-Vu conjecture.

arxiv 2009.08846 v2 pith:IKDPJNAP submitted 2020-09-18 math.CO math.NT

Cosmin Pohoata , Dmitriy Zakharov This is my paper · ORCID
classification math.COmath.NT
keywords mathbbmathcalsubsetconstantepsilonolsonadditivecardinality
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The pith

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The reading

Additive combinatorics asks how many elements from a group you need before some nonempty collection of them adds to zero. For the field with p elements, this number is about the square root of p; for d-dimensional vector spaces it was conjectured to be about (d-1)p. This paper proves that upper bound. More precisely, for any fixed d and any tolerance epsilon, once p is large, every subset with more than (d-1+epsilon)p vectors contains a nonempty zero-sum subcollection.

Previous progress stopped at dimensions two and three. The proof decomposes a large set into structured pieces, projects the pieces onto a lower-dimensional space, uses the polynomial method to find a zero-sum combination in the projection, and then lifts that combination back to an actual subset of the original vectors. A probabilistic step ensures the lifted pieces are spread out enough for the lifting to work. The argument borrows and adapts machinery from the second author's work on the Erdos-Ginzburg-Ziv problem.

The theorem is asymptotic: it does not give explicit bounds on p, and it does not compute the Olson constant exactly. The value is in confirming the conjectured linear dependence on dimension and in providing a framework that may be useful for other zero-sum questions.

Extended reading notes

Core claim

Theorem 1: For any fixed d>=2 and epsilon>0, the Olson constant of F_p^d satisfies OL(F_p^d) <= (d-1+epsilon)p for all sufficiently large primes p. If true, this gives the conjectured asymptotic threshold for zero-sum-free subsets: any set of vectors exceeding about (d-1)p elements must contain a nonempty subset summing to zero.

Load-bearing premise

The main theorem depends on the iterative expansion step in Proposition 1, which uses Lemma 4. The second inequality of Lemma 4 is not proved in this paper; the text says 'The second inequality follows from [22, Lemma 3.1]', a result in the second author's earlier preprint. If that expansion estimate is false under the hypotheses used for the difference set X-X (l=0 case) or for the auxiliary function f (l>0 case), the lifting construction fails and the theorem does not follow.

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Formalized claims in Lean

  1. Claim #1: Theorem 1: For any fixed d>=2 and epsilon>0, the Olson constant of F_p^d satisfies OL(F_p^d) <= (d-1+epsilon)p for all sufficiently large primes p. If true, this gives the conjectured asymptotic threshold for zero-sum-free subsets: any set of vectors exceeding about (d-1)p elements must contain a nonempty subset summing to zero.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: epsilon is part of the theorem statement, and K0, g, K2, mu, delta are auxiliary constants chosen by the proof. The central argument relies on standard theorems (Combinatorial Nullstellensatz, Loomis-Whitney, Cauchy-Davenport, Chernoff) and imports one nontrivial expansion lemma from the second author's earlier paper [22]. No new entities are postulated.

assumptions (5)
  • standard math Combinatorial Nullstellensatz (Alon's theorem)
    Used in Lemma 7 to produce a non-zero coefficient vector with vanishing sum.
  • standard math Loomis-Whitney inequality
    Used in Lemma 4 to lower-bound the expansion of a set under translation by a basis vector.
  • standard math Cauchy-Davenport theorem in F_p^d
    Used in Proposition 1 to show that two sumsets of size greater than p^{d/2} cover the group.
  • standard math Chernoff bound for binomial random variables
    Used in Proposition 2 to show random subsets Z_y are thick with high probability.
  • domain assumption Expansion inequality from [22, Lemma 3.1]
    The second inequality in Lemma 4 is imported without proof from the second author's prior paper; it is essential to the iterative growth step in Proposition 1.

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Pith. "Pith review of Zero subsums in vector spaces over finite fields." pith.science (2026). https://pith.science/paper/IKDPJNAP

@misc{pith2026200908846,
  author       = {Pith},
  title        = {Pith review of: Zero subsums in vector spaces over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKDPJNAP}},
  note         = {Machine review of arXiv:2009.08846}
}
abstract

The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\subset \mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{O}L(\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d \geq 2$ and $\epsilon > 0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality $$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+\epsilon)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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