On the size of subsets of $\mathbb{F}_p^{n}$ without $p$ distinct elements summing to zero

Sauermann, Lisa

doi:10.48550/arxiv.1904.09560

Cited by 4 publications

(19 citation statements)

References 15 publications

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“…Even in the case where (⋆) consists only of one equation (i.e. in the case m = 1) this is a widely open problem, and in the special case of the equation x 1 + • • • + x p = 0 it has applications to bounding Erdős-Ginzburg-Ziv constants (see [9,21,24]).…”

Section: Discussionmentioning

confidence: 99%

“…In the case of the single equation x 1 + • • • + x p = 0, which is very closely related to the Erdős-Ginzburg-Ziv problem in discrete geometry, Naslund [21] proved a bound of the form |A| ≤ C p • (Γ p ) n for some Γ p between 0.84p and 0.92p (the constant C p was later improved in [9]). The best current bound in this case is |A| ≤ C p • (2 √ p) n due to the author [24]. In the case of a general single linear equation a…”

Section: Introductionmentioning

confidence: 99%

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Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$

Sauermann¹

2021

Preprint

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Let us fix a prime p and a homogeneous system of m linear equations aj,1x1 + • • • + a j,k x k = 0 for j = 1, . . . , m with coefficients aj,i ∈ Fp. Suppose that k ≥ 3m, that aj,1 + • • • + a j,k = 0 for j = 1, . . . , m and that every m × m minor of the m × k matrix (aj,i)j,i is non-singular. Then we prove that for any (large) n, any subset A ⊆ F n p of size |A| > C • Γ n contains a solution (x1, . . . , x k ) ∈ A k to the given system of equations such that the vectors x1, . . . , x k ∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ < p.The crucial point here is the condition for the vectors x1, . . . , x k in the solution (x1, . . . , x k ) ∈ A k to be distinct. If we relax this condition and only demand that x1, . . . , x k are not all equal, then the statement would follow easily from Tao's slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$

Sauermann¹

2021

Preprint

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“…Let us note that the opposite case when p is fixed and d is large is also of great interest. The current best bounds are s(F d 3 ) 2.756 d proved by Ellenberg-Gijswijt in their breakthrough paper [6] and s(F d p ) C p (2 √ p) d for p 5 due to Sauermann [17]. See [17] and references therein for an exposition of the existing results.…”

Section: Introductionmentioning

confidence: 99%

“…The current best bounds are s(F d 3 ) 2.756 d proved by Ellenberg-Gijswijt in their breakthrough paper [6] and s(F d p ) C p (2 √ p) d for p 5 due to Sauermann [17]. See [17] and references therein for an exposition of the existing results. Let us remark that the case p = 3 is also related to the famous cap set problem: 1 2 s(F d 3 ) equals to the maximal cardinality of a set A ⊂ F d 3 such that A does not contain any affine lines.…”

Section: Introductionmentioning

confidence: 99%

Convex geometry and Erdős-Ginzburg-Ziv problem

Zakharov¹

2020

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Denote by s(F d p ) the minimal number s such that among any s (not necessarily distinct) vectors in F d p one can find p vectors whose sum is zero. Denote by w(F d p ) the weak Erdős-Ginzburg-Ziv constant, that is, the maximal number of vectors v 1 , . . . , v s ∈ F d p such that for any non-negative integers α 1 , . . . , α s whose sum is p we have α 1 v 1 + . . . + α s v s = 0 if and only if α i = p for some i. We show that for any p and d we have an upper bound w(F d p )2d−1 d + 1. The main result of this paper is that for any fixed d and p → ∞ we have an asymptotic formula s(F d p ) ∼ w(F d p )p. Together with the upper bound on w(F d p ) this result in particular implies that s(F d p ) 4 d p for all sufficiently large p. In order to prove the main result, we develop a framework of convex flags which generalize usual polytopes in many ways. Many classical results of Convex Geometry translate naturally to this new setting. In particular, we obtain analogues of Helly Theorem and of Central Point Theorem. Also we prove a generalization of Integer Helly Theorem of Doignon. One of the main tools in our argument is the Flag Decomposition Lemma which asserts that for any subset X ⊂ F d p one can find a convex flag which approximates X in a certain way. Then, Integer Central Point Theorem and other tools allow us to solve the problem for this approximation. Finally, in order to lift the solution back to the original set X we apply the Set Expansion method of Alon-Dubiner.

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“…, x 2k+1 ) ∈ A 2k+1 of S * k . (Semishapes correspond to cycles in [Sau19].) We say it is degenerate if #{x 1 , x 2 , .…”

Section: Introductionmentioning

confidence: 99%

Avoiding a star of three-term arthmetic progressions

Mimura,

Tokushige

2019

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On the size of subsets of $\mathbb{F}_p^{n}$ without $p$ distinct elements summing to zero

Cited by 4 publications

References 15 publications

Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$

Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$

Convex geometry and Erdős-Ginzburg-Ziv problem

Avoiding a star of three-term arthmetic progressions

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