Growth Estimates in Positive Characteristic via Collisions

Yazici, Esen Aksoy; Murphy, Brendan; Rudnev, Misha; Shkredov, Ilya D.

doi:10.1093/imrn/rnw206

Cited by 34 publications

(87 citation statements)

References 27 publications

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“…Theorem 1 has recently found many applications in sum-product type estimates in, e.g., [38], [3], [33] where the arising sets of points and planes have natural structure of Cartesian products. In particular, in [3,Corollary 6], it was observed that Theorem 1 implied a pointline incidence bound in F 2 in the special case of the point set being a Cartesian product.…”

Section: Other Statements Of Theorem 1 and Point-line Incidence Boundmentioning

confidence: 99%

11. Point-plane incidences and some applications in positive characteristic

Rudnev¹

2019

Combinatorics and Finite Fields

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The point-plane incidence theorem states that the number of incidences between n points and m ≥ n planes in the projective three-space over a field F , iswhere k is the maximum number of collinear points, with the extra condition n < p 2 if F has characteristic p > 0. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in F 2 , due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in F d , for d = 2, 3, 4. These are the problem of the minimum number of distinct nonzero values of a non-degenerate bilinear form on a point set in d = 2, the analogue of the Erdős distinct distance problem in d = 2, 3 and additive energy estimates for sets, supported on a paraboloid and sphere in d = 3, 4. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.2000 Mathematics Subject Classification. 68R05,11B75.1 the folklore that the proof should work, with some constraints, over a general field. The first "official" account of this was given by Ellenberg and Hablicsek [16] in late 2013, followed by Kollár [28] and the author [39] in 2014. The latter two had been aware of a 2003 paper by Voloch [50], which discussed the constraints under which the key element of the First Guth-Katz theorem proof, the Monge-Salmon theorem [42], applied in positive characteristic.Theorem 2 (First Guth-Katz theorem). Let L be a set of lines in C 3 . Suppose, no more then two lines are concurrent. Then the number of pair-wise intersections of lines in L is O |L| 3 2 + |L|k ,

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Section: Other Statements Of Theorem 1 and Point-line Incidence Boundmentioning

confidence: 99%

11. Point-plane incidences and some applications in positive characteristic

Rudnev¹

2019

Combinatorics and Finite Fields

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“…where τ = τ (δ, K) = δ 2 (log 2K) −3+o (1) . Whence for all sufficiently large p, namely, for log p/ log log p ≫ δ −2 (log 2K) 3+o(1) ,…”

Section: The Proofs Of the Main Resultsmentioning

confidence: 99%

“…Using the method of the proof of Theorem 1 as well as some last results from sum product theory [1], we obtain an upper bound for the ternary sum in the case of sets with small additive doubling. …”

Section: Introductionmentioning

confidence: 99%

Sums of multiplicative characters with additive convolutions

Volostnov

Shkredov

2017

Proc. Steklov Inst. Math.

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“…For d = 2, making a linear changing of the variables one can assume that f (Z) = αZ 2 + β ∈ F p [Z] with α = 0. Then, clearly, the quantity T + 3 (f (A)) is equal to the number of solutions to f (a 1 ) + 2αb 1…”

Section: 4mentioning

confidence: 99%

“…provided that for some fixed real ξ > 0 and ζ > 0 we have S = p ξ+o(1) , X = p ζ+o (1) and for k = ζ −1 we have (1.2) ξ > 3k − 2 − 4kζ 6k − 8 (where δ > 0 depends only on ξ and ζ).…”

Section: Introductionmentioning

confidence: 99%