Fp is locally like ℂ

Grosu, Codruţ

doi:10.1112/jlms/jdu007

Cited by 16 publications

(19 citation statements)

References 33 publications

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“…Over finite fields, the extremal situations of ‘very small’ or ‘very large’ sets are relatively well understood (where size is relative to the cardinality of the field). For very small sets, Grosu achieved the optimal bound

O (N^{4 / 3})

over

F_{p}

, if

m, n ⩽ N

and

5 N < {prefixlog}_{2} {prefixlog}_{6} {prefixlog}_{18} p

. For very large sets, Vinh proved

I false(scriptP, scriptL false) ⩽ m n / q + q^{1 / 2} \sqrt{m n}

over any finite field

F_{q}

.…”

Section: Introductionmentioning

confidence: 99%

An improved point-line incidence bound over arbitrary fields

Stevens

Zeeuw

2017

Bull. London Math. Soc.

112

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Abstract. We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F 2 , with m 7/8 < n < m 8/7 , determine at most O(m 11/15 n 11/15 ) incidences (where, if F has positive characteristic p, we assume m −2 n 13 ≪ p 15 ). This improves on the previous best known bound, due to Jones.To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned.We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.

show abstract

O (N^{4 / 3})

over

F_{p}

, if

m, n ⩽ N

and

5 N < {prefixlog}_{2} {prefixlog}_{6} {prefixlog}_{18} p

. For very large sets, Vinh proved

I false(scriptP, scriptL false) ⩽ m n / q + q^{1 / 2} \sqrt{m n}

over any finite field

F_{q}

.…”

Section: Introductionmentioning

confidence: 99%

An improved point-line incidence bound over arbitrary fields

Stevens

Zeeuw

2017

Bull. London Math. Soc.

112

View full text Add to dashboard Cite

show abstract

“…It is worth noting that it follows from results of Grosu [7] and of Tao [18] that the minrank of a graph G over C is a lower bound for its minrank over every field F whose characteristic is sufficiently large as a function of G. By combining these results with Theorem 1.2, we immediately get that for every n −1 ≤ p ≤ 1, the random graph G = G(n, p) w.h.p. satisfies min-rank F (G) ≥ Ω n log(1/p) log n for every field F of characteristic which is sufficiently large as a function of n. The stronger assertion of Theorem 5.1 follows by replacing the result of [17] (Lemma 2.3) by that of [16] in the proof of Theorem 1.2.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 76%

The minrank of random graphs over arbitrary fields

et al. 2019

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The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M ∈ F n×n with nonzero diagonal entries such that M i,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field F = F(n) and every p = p(n) satisfying n −1 ≤ p ≤ 1 − n −0.99 , the minrank of G = G(n, p) over F is Θ( n log(1/p) log n ) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most n O(1) , with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.

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“…Heuristically, if Conjecture 1 were true over F p , it might be possible to make an expanding family of Schreier graphs for Aff (1, F p ) F p with bounded degree (following a similar strategy to Bourgain and Gamburd [6]), however this is known to be false by a theorem of Lubotzky and Weiss [27,26]. To prove the lower bound in Theorem 3, we use a construction of Klawe [24,25], which gives a quantitative proof of Lubotzky and Weiss' theorem for Aff(1, F p ) F p ; using a theorem of Grosu [20], we embed our counter-example into C 2 .…”

Section: Rlgp(fnα) ≤ C(α)mentioning

confidence: 99%

“…Thus to ensure (log log q)q (3+δ)s p, it suffices to take |Y | ≤ p 1−ε for any ε > 0. The proof of Theorem 19 is an application of a rectification theorem of Grosu [20], which allows us to embed small subsets of F p into C while preserving algebraic equations of low complexity. In particular, Grosu's theorem allows up to embed the counterexamples constructed in Theorem 15 into C 2 .…”

Section: Quantitative Lower Bounds Over Rmentioning

confidence: 99%

Upper and lower bounds for rich lines in grids

Murphy¹

2021

American Journal of Mathematics

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We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby. The upper bounds are based on a version of the asymmetric Balog-Szemerédi-Gowers theorem for group actions combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and amenability (or expanding families of graphs in the finite field case). As an application of our upper bounds for rich lines in grids, we give a geometric proof of the asymmetric sum-product estimates of Bourgain and Shkredov. Contents. 1. Introduction. 2. Lower bounds for rich lines in grids. 3. Upper bounds for rich lines in grids. Appendix A. Proof of group action Balog-Szemerédi-Gowers. Appendix B. Product theorems for Aff(1, F). References. 1. Introduction. Let F be a field and let 0 < α ≤ 1 be a real number. A line in the plane F 2 is α-rich in a Cartesian product point set Y ×Y ⊆ F 2 if ∩ (Y × Y) ≥ α|Y |. For short, we call Y × Y a grid. Any line contains at most N points of a N × N grid, so a line is α-rich if it contains α-percent of the maximum possible points of incidence. The parameter α may be a constant independent of N , or may be some small power of 1/N. There are two questions we wish to answer about rich lines in grids: (1) how many α-rich lines can a N × N grid support? (2) if a grid supports many rich lines, must these lines have some structure? The first question was answered for F = R by Szemerédi-Trotter [37]: a N × N grid has at most O(α −3 N) α-rich lines; this is sharp. Below, we discuss extensions of this upper bound and the examples that provide lower bounds. (We use standard asymptotic notation; see Section 1.3 for definitions.

show abstract

Fp is locally like ℂ

Cited by 16 publications

References 33 publications

An improved point-line incidence bound over arbitrary fields

An improved point-line incidence bound over arbitrary fields

The minrank of random graphs over arbitrary fields

Upper and lower bounds for rich lines in grids

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