Distinct spreads in vector spaces over finite fields

Lund, Ben; Pham, Thang; Vinh, Lê Anh

doi:10.1016/j.dam.2017.12.027

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…(The result from this theorem has been improved in [12]. They show that |E| ≥ (1 + ǫ)q ⌈d/2⌉ is sufficient to guarantee cq spreads and that this estimate is tight.…”

Section: Spreadmentioning

confidence: 99%

Occurrence of right angles in vector spaces over finite fields

Bennett

2018

European Journal of Combinatorics

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Here we examine some Erdős-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A ⊂ F d q of size ≥ cq d+2 3 contains three points which generate a right angle, and b) a subset A ⊂ F d q of size ≥ Cq d+2 2contains two points which generate a right angle with the vertex at the origin. We will also prove that b) is sharp up to constants and provide some partial results for similar problems related to spread and collinear triples.Theorem 2.2. If A, B ⊂ F d q , A ∩ B = ∅, and |A| 2 |B| ≥ 4q d+2 , then there are x, y ∈ A and z ∈ B so that (x, y, z) forms a right angle.Following immediately from the theorem, we have:

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“…(The result from this theorem has been improved in [12]. They show that |E| ≥ (1 + ǫ)q ⌈d/2⌉ is sufficient to guarantee cq spreads and that this estimate is tight.…”

Section: Spreadmentioning

confidence: 99%

Occurrence of right angles in vector spaces over finite fields

Bennett

2018

European Journal of Combinatorics

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“…In [11], Pham, Phuong, Sang, Valculescu, and Vinh consider distances between points and lines, and prove that if P and L are sets of points and lines, respectively, with |P | • |L| sufficiently large, then one obtains a positive proportion of distances. In [9], Lund, Pham, and Vinh study the problem where distance is replaced by a finite field analogue of angle(defined by dividing the dot product by the lengths, in analogue to the usual geometric formula for the sine of the angle between vectors). The authors obtain a non-trivial exponent which ensures a positive proportion of these angles are obtained.…”

Section: Introductionmentioning

confidence: 99%

Congruence classes of large configurations in vector spaces over finite fields

McDonald¹

2020

Funct. Approx. Comment. Math.

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In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent s < d such that any set E ⊂ F d q with |E| q s determines q ( k+1 2 ) congruence classes of (k + 1)-point configurations for k ≤ d. Because congruence classes can be identified with tuples of distances between distinct points when k ≤ d, and because there are k+1 2 such pairs, this means any such E determines a positive proportion of all congruence classes. In the k > d case, fixing all pairs of distnaces leads to an overdetermined system, so q ( k+1 2 ) is no longer the correct number of congruence classes. We determine the correct number, and prove that |E| q s still determines a positive proportion of all congruence classes, for the same s as in the k ≤ d case.

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“…Another interesting variant of this problem was studied in [10], where Lund, Pham, and Vinh defined the angle between two vectors in analogue with the usual geometric interpretation of the dot product. Namely, given vectors x and y, they consider the quantity…”

Section: Introductionmentioning

confidence: 99%

Areas of Triangles and SL2 Actions in Finite Rings

McDonald¹

2019

bulmathenu

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In Euclidean space, one can use the dot product to give a formula for the area of a triangle in terms of the coordinates of each vertex. Since this formula involves only addition, subtraction, and multiplication, it can be used as a definition of area in R 2 , where R is an arbitrary ring. The result is a quantity associated with triples of points which is still invariant under the action of SL2(R). One can then look at a configuration of points in R 2 in terms of the triangles determined by pairs of points and the origin, considering two such configurations to be of the same type if corresponding pairs of points determine the same areas. In this paper we consider the cases R = Fq and R = Z/p ℓ Z, and prove that sufficiently large subsets of R 2 must produce a positive proportion of all such types of configurations.

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Distinct spreads in vector spaces over finite fields

Cited by 3 publications

References 24 publications

Occurrence of right angles in vector spaces over finite fields

Occurrence of right angles in vector spaces over finite fields

Congruence classes of large configurations in vector spaces over finite fields

Areas of Triangles and SL2 Actions in Finite Rings

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