Bisectors and Pinned Distances

Lund, Ben; Petridis, Giorgis

doi:10.1007/s00454-019-00122-w

Cited by 7 publications

(21 citation statements)

References 21 publications

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“…They proved that for E ⊂ F 2 p with p ≡ 3 mod 4, if |E| ≪ p . This result has been recently improved by Lund and Petridis in [9], namely, they indicated that |∆(E)| |E| 1 2 + 3 74 , under the condition that |E| ≪ p 8/5 with p ≡ 3 mod 4.…”

Section: Introductionmentioning

confidence: 80%

New bounds for distance-type problems over prime fields

Iosevich

Koh

Pham

2020

European Journal of Combinatorics

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Section: Introductionmentioning

confidence: 80%

New bounds for distance-type problems over prime fields

Iosevich

Koh

Pham

2020

European Journal of Combinatorics

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“…We apply Theorem 7 to 𝑃 and Π, claiming that the number of collinear points is bounded by 𝑀. This claim is a direct consequence of [22,Lemma 2.2]. Firstly, note that points in 𝑃 ∶= 𝜅(𝐺 ′ 𝑟 ) correspond to a segment 𝑠 𝑟 ∈ 𝑆 𝑟 in the configuration space; 𝑠 𝑟 determines the rigid motion g via the relation g −1 (𝜏(𝑠 𝑟 )) = (𝑎, 𝑎 ′ ) and 𝜅(g) is a point in 𝑃.…”

Section: Proof Of Theorem 11mentioning

confidence: 90%

“…If 𝑏 ≠ 𝑏 ′ , then it's easy to conclude that 𝑑(𝑎, 𝑏) = 𝑑(𝑎, 𝑏 ′ ) = 0 (see, e.g. [22,Lemma 2.3]). This is a contradiction, so there are |𝐴| 2 such triples.…”

Section: Isosceles Triangles and Bisector Energymentioning

confidence: 99%

On the pinned distances problem in positive characteristic

Murphy

Petridis

Pham

et al. 2022

Journal of London Math Soc

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We study the Erdős-Falconer distance problem for a set 𝐴 ⊂ 𝔽 2 , where 𝔽 is a field of positive characteristic 𝑝. If 𝔽 = 𝔽 𝑝 and the cardinality |𝐴| exceeds 𝑝 5∕4 , we prove that 𝐴 determines an asymptotically full proportion of the feasible 𝑝 distances. For small sets 𝐴, namely when |𝐴| ⩽ 𝑝 4∕3 over any 𝔽, we prove that either 𝐴 determines ≫ |𝐴| 2∕3 distances, or 𝐴 lies on an isotropic line. For both large and small sets, the results proved are in fact for pinned distances. 2 0 2 0 ) 28A75, 52C10 (primary), 11B30 (secondary) M S C ( INTRODUCTIONFor a compact set 𝐴 ⊆ ℝ 2 , the Falconer conjecture [12] claims that, for the distance set of 𝐴 to have positive Lebesgue measure, the Hausdorff dimension 𝑑 𝐻 of 𝐴 must be strictly greater than 1.Recently, Guth et al.[15] made significant progress by establishing the inequality 𝑑 𝐻 ⩾ 5∕4 using decoupling. This improved the estimate 𝑑 𝐻 ⩾ 4∕3 due to Wolff [38] some 20 years previously. Iosevich and the fourth author [19] stated a finite field 𝔽 𝑞 -variant of the Falconer conjecture. The Wolff exponent 4∕3 was subsequently proved by Chapman et al. [8] for 𝑞 ≡ 3 mod (4); this constraint was removed by Bennett et al. [2]. Here, we define the distance 𝑑 between two points 𝑥 = (𝑥 1 , 𝑥 2 ) and 𝑦 = (𝑦 1 , 𝑦 2 ) in the plane 𝔽 2 𝑞 to be(1)These approaches were based on Fourier analysis, with the underlying considerations similar to those used over the reals. In contrast, there is no positive characteristic analogue of the multiscale

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“…The best current lower bound is |E| 2/3 due to Murphy, Petridis, Pham, Rudnev and Stevens [19]. We also note that it seems very difficult to extend the methods in [12,16,11,19] to the Minkowski and parabola distance functions. We now take advantage of the generality of Theorem 1.1 to study algebraic distances between two sets in F 3 p , where one set lies on a plane and the other set is arbitrary.…”

Section: Pinned Algebraic Distancesmentioning

confidence: 95%

“…When f (x, y) = x 2 + y 2 , Theorem 6.1 was first proved by Stevens and de Zeeuw in [24] by using a point-line incidence bound. The exponent 8 15 was improved to 1 2 + 149 4214 by Iosevich, Koh, Pham, Shen and Vinh [12], then to 1 2 + 3 74 by Lund and Petridis [16] and to 1 2 + 69 1558 by Iosevich, Koh and Pham [11]. The best current lower bound is |E| 2/3 due to Murphy, Petridis, Pham, Rudnev and Stevens [19].…”

Section: Pinned Algebraic Distancesmentioning

confidence: 98%

A Point-Conic Incidence Bound and Applications over $\mathbb F_p$

Mohammadi¹,

Pham²,

Warren³

2021

Preprint

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In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.

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Bisectors and Pinned Distances

Abstract: We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their bisector energy.

Cited by 7 publications

References 21 publications

New bounds for distance-type problems over prime fields

New bounds for distance-type problems over prime fields

On the pinned distances problem in positive characteristic

A Point-Conic Incidence Bound and Applications over $\mathbb F_p$

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