The Szemerédi–Trotter type theorem and the sum-product estimate in finite fields

Vinh, Lê Anh

doi:10.1016/j.ejc.2011.06.008

Cited by 108 publications

(121 citation statements)

References 11 publications

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“…The above result was also proved for points and hyperplanes, and for points and k-subspaces (see [4,20] for more details).…”

Section: P||l|mentioning

confidence: 87%

“…Let P be a set of points, L a set of lines over F d q , and I(P, L) the number of incidences between P and L. Bourgain, Katz, and Tao [5] proved that for any 0 < α < 2 and |P |, |L| ≤ N = q α , I(P, L) N 3/2−ǫ , where ǫ = ǫ(α). By employing the Erdős-Rényi graph (see 2.1 for the definition), the third author [20] improved this bound in the case 1 ≤ α ≤ 2, and gave the following estimate. …”

Section: Introductionmentioning

confidence: 99%

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Incidences between points and generalized spheres over finite fields and related problems

2016

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Let F q be a finite field of q elements where q is a large odd prime power andWe prove bounds on the number of incidences between a point set P and a Q-sphere set S, denoted by I(P, S), as the following.We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings Z q where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections 4 and 5, we prove a bound on the number of incidences between a random point set and a random Q-sphere set in F d q . We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

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“…The above result was also proved for points and hyperplanes, and for points and k-subspaces (see [4,20] for more details).…”

Section: P||l|mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Incidences between points and generalized spheres over finite fields and related problems

2016

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“…In his elegant paper, Vu relates the sum-product bound to the expansion of certain graphs, and then via the relation of the spectrum (second eigenvalue) and expansion one can deduce a rather strong bound. Vinh [336] (also see [337]), using ideas from spectral graph theory, derived a Szemerédi-Trotter type theorem in finite fields, and from there obtained a different proof of Garaev's result [128] on sum-product estimate for large subsets of finite fields. Also, Solymosi [294] applied techniques from spectral graph theory and obtained estimates similar to those of Garaev [126] that already followed via tools from exponential sums and Fourier analysis.…”

Section: A|mentioning

confidence: 99%

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

Bibak

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define -perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

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“…For more details on this result and for other results on the distance graph of F d q , see [1], [3], [6] and [7].…”

Section: Introductionmentioning

confidence: 99%

Long Paths in the Distance Graph Over Large Subsets of Vector Spaces Over Finite Fields

Bennett¹,

Chapman²,

Covert³

et al. 2016

Journal of the Korean Mathematical Society

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Abstract. Let E ⊂ F d q , the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y ∈ E by an edge if ||x − y|| := (We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, 1 · |E| k+1 q −k plus a much smaller remainder.

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The Szemerédi–Trotter type theorem and the sum-product estimate in finite fields

Abstract: We study a Szemerédi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

Cited by 108 publications

References 11 publications

Incidences between points and generalized spheres over finite fields and related problems

Incidences between points and generalized spheres over finite fields and related problems

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

Long Paths in the Distance Graph Over Large Subsets of Vector Spaces Over Finite Fields

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