Repeated Angles in Three and Four Dimensions

Apfelbaum, Roel; Sharir, Micha

doi:10.1137/s0895480104443941

Cited by 8 publications

(29 citation statements)

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“…This simple observation justifies the various (entirely routine) localization arguments that follow, as well as the interplay between Fourier transforms of functions and Fourier transforms of measures supported on P n−1 . In particular let K (1) p,n (δ) be the smallest constant such that…”

Section: Linear Versus Multilinear Decouplingmentioning

confidence: 99%

The proof of the l^2 Decoupling Conjecture

2015

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Abstract. We prove the l 2 Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which implies the full range of expected L p x,t Strichartz estimates for both the rational and (up to N ǫ losses) the irrational torus. Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere. Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory. The l 2 Decoupling TheoremLet S be a compact C 2 hypersurface in R n with positive definite second fundamental form. Examples include the sphere S n−1 and the truncated (elliptic) paraboloidUnless specified otherwise, we will implicitly assume throughout the whole paper that n ≥ 2. We will write A ∼ B if A B and B A. The implicit constants hidden inside the symbols and ∼ will in general depend on fixed parameters such as p, n and sometimes on variable parameters such as ǫ, ν. We will not record the dependence on the fixed parameters.Let N δ be the δ neighborhood of P n−1 and let P δ be a finitely overlapping cover of N δ with curved regions θ of the formwhere C θ runs over all cubes cIt is also important to realize that the normals to these boxes are ∼ δ 1/2 separated. A similar decomposition exists for any S as above and we will use the same notation P δ for it. We will denote by f θ the Fourier restriction of f to θ.Our main result is the proof of the following l 2 Decoupling Theorem.Key words and phrases. discrete restriction estimates, Strichartz estimates, additive energy. The first author is partially supported by the NSF grant DMS-1301619. The second author is partially supported by the NSF Grant DMS-1161752. and ǫ > 0Theorem 1.1 has been proved in [20] for p > 2 +. A standard construction is presented in [20] to show that, up to the δ −ǫ term, the exponent of δ is optimal. We point out that Wolff [36] has initiated the study of l p decouplings, p > 2 in the case of the cone. His work provides part of the inspiration for our paper.A localization argument and interpolation between p = 2(n+1) n−1 and the trivial bound for p = 2 proves the subcritical estimate. Estimate (3) is false for p < 2. This can easily be seen by testing it with functions of the form f θ (x) = g θ (x + c θ ), where supp( g θ ) ⊂ θ and the numbers c θ are very far apart from each other.Inequality (3) has been recently proved by the first author for p = We mention briefly that there is a stronger form of decoupling, sometimes referred to as square function estimate, which predicts thatin the slightly smaller range 2 ≤ p ≤ 2n n−1. When n = 2 this easily follows via a geometric argument. Minkowski's inequality shows that (4) is indeed stronger than (3) in the range 2 ≤ p ≤ 2n n−1 . This is also confirmed by the lack of any results for (4) when n...

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Section: Linear Versus Multilinear Decouplingmentioning

confidence: 99%

The proof of the l^2 Decoupling Conjecture

2015

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“…Formally speaking, suppose, there is a finite set of lines L * in P 3 . Define the restricted set of incidences between a point set Q and set of planes Π as (2) I * (Q, Π) = {(q, π) ∈ Q × Π : q ∈ π and ∀l ∈ L * , q ∈ l or l ⊂ π}.…”

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

confidence: 99%

“…After that we may assume that S 2 t contain no lines and apply Theorem 4 in a way the Szemerédi-Trotter theorem was used by Appelbaum and Sharir [2]. One fixes z ∈ A and counts the maximum number of right triangles in A with the vertex z.…”

mentioning

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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“…This is a relatively straight forward generalization of the argument of Apfelbaum and Sharir in [1]. Assume for simplicity that n is a d-th power and a multiple of 5 so that all the quantities in the proof are integers.…”

Section: Proof Of Theorem 112mentioning

confidence: 92%

“…For every d ≥ 2 and s ∈ (0, d 2 ), there exists E ⊂ R d of Hausdorff dimension s such that π 2 is not equitably represented in A(E). The main ingredient in the proof is the following generalization to R d of a theorem by Apfelbaum and Sharir in [1], which they state in R 3 .…”

Section: Introductionmentioning

confidence: 99%

On angles determined by fractal subsets of the Euclidean space

Iosevich

Mourgoglou

Palsson

2016

Mathematical Research Letters

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We prove that if the Hausdorff dimension of a compact subset of R d is greater than d+1 2 , then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in ([8]). We also obtain new upper bounds for the number of times an angle can occur among N points in R d , d ≥ 4, motivated by the results of Apfelbaum and Sharir ([1]) and Pach and Sharir ([13]). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.

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Repeated Angles in Three and Four Dimensions

Cited by 8 publications

References 6 publications

The proof of the l^2 Decoupling Conjecture

The proof of the l^2 Decoupling Conjecture

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

On angles determined by fractal subsets of the Euclidean space

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