New restriction estimates for the 3-d paraboloid over finite fields

Lewko, Mark

doi:10.1016/j.aim.2014.11.008

Cited by 31 publications

(46 citation statements)

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“…In [8], [4] after applying the Hölder inequality the spherical average estimate was converted to an additive energy estimate on the sphere (which was trivial as the sphere was S 1 t ). It is easy to show -see (18) and (29) below -that if the set A lies on the discrete paraboloid or sphere, then (up to a permutation of vertices) the energy equals, respectively, the number of rectangles formed in F d−1 by the horizontal projections of the points x, y, z, u on the paraboloid or the rectangles with vertices x, y, z, u on the sphere in F d . We always assume that x, y, z, u in (16) are pair-wise distinct, for alternative scenarios contribute merely O(|A| 2 ) to the energy.…”

Section: Additive Energy On Quadricsmentioning

confidence: 99%

“…We always assume that x, y, z, u in (16) are pair-wise distinct, for alternative scenarios contribute merely O(|A| 2 ) to the energy. By a rectangle we mean a point quadruple, such that the dot products of adjacent difference vectors is zero at every vertex x, y, z, u -see (18), (29). The concept of adjacent vertices arises after rearranging equation (16).…”

Section: Additive Energy On Quadricsmentioning

confidence: 99%

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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: Additive Energy On Quadricsmentioning

confidence: 99%

Section: Additive Energy On Quadricsmentioning

confidence: 99%

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

Rudnev¹

2019

Combinatorics and Finite Fields

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“…In even dimensions the Stein-Tomas result was improved by the first two authors [8] in 2008 to R * (2 → r) 1 for r > 2d 2 d 2 −2d+2 , with the endpoint r = 2d 2 d 2 −2d+2 obtained in [15]. The second author [12] recently improved this to r > 6d+8 3d−2 for d ≥ 6, using, in part, ideas from [13]. Here we further improve these results in d ≥ 4 obtaining the optimal L 2 based results for even d ≥ 8.…”

Section: Introductionmentioning

confidence: 95%

Finite field restriction estimates for the paraboloid in high even dimensions

Iosevich

Koh

Lewko

2020

Journal of Functional Analysis

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We prove that the finite field Fourier extension operator for the paraboloid is bounded from L 2 → L r for r ≥ 2d+4 d in even dimensions d ≥ 8, which is the optimal L 2 estimate. For d = 6 we obtain the optimal range r > 2d+4 d = 8/3, apart from the endpoint. For d = 4 we improve the prior range of r > 16/5 = 3.2 to r ≥ 28/9 = 3.111 . . ., compared to the conjectured range of r ≥ 3.The key new ingredient is improved additive energy estimates for subsets of the paraboloid.2010 Mathematics Subject Classification. 42B05.

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“…In the paraboloid case, we will see that (proof of Theorem 1.16), there is a connection between the L 2 → L r estimate and the additive energy bound. In the finite field setting, such a connection was initially given by Mockenhaupt and Tao [42], and a more precise relation between them was found by Lewko [44]. However, it seems that there is no such link for the case of spheres.…”

Section: Extension Theoremsmentioning

confidence: 98%

“…This result has been improved slightly over recent years. For instance, Lewko [44] showed that there exists ǫ > 0 such that…”

Section: Extension Theoremsmentioning

confidence: 99%

Extension theorems in vector spaces over finite fields

Koh¹

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In this paper, we break the exponent (d + 1)/2 for some distance problems in the spaces over arbitrary finite fields. We also obtain new extension theorems for paraboloids and spheres in odd dimensions. Our L 2 → L r extension theorems for paraboloids in odd dimensions improve significantly the exponent obtained by the first listed author. Our L p → L 4 extension theorems for spheres in odd dimensions break the Stein-Tomas result toward L p → L 4 which has stood for more than ten years. It follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the L p − L 4 estimates for spheres with primitive radii are much stronger than the optimal results for paraboloids given by Mockenhaupt and Tao. We will provide connections between the Erdős-Falconer distance problem over finite fields and extension conjectures for paraboloids and spheres. In addition, we also consider the problem of product of simplices in F d q and related topics. Our method is a combination of techniques from harmonic analysis over finite fields, methods from spectral graph theory, and some tools from group action theory and algebraic combinatorics. 1128 2107 under the assumption |A| ≪ q 7/6 due to Iosevich et al. [33].

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New restriction estimates for the 3-d paraboloid over finite fields

Cited by 31 publications

References 18 publications

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

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

Finite field restriction estimates for the paraboloid in high even dimensions

Extension theorems in vector spaces over finite fields

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