2018
DOI: 10.1016/j.dam.2017.12.027
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Distinct spreads in vector spaces over finite fields

Abstract: In this short note, we study the distribution of spreads in a point set P ⊆ F d q , which are analogous to angles in Euclidean space. More precisely, we prove that, for any ε > 0, if |P| ≥ (1 + ε)q ⌈d/2⌉ , then P determines a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets P ⊂ F d q of size |P| = q ⌈d/2⌉ that determine at most one spread.1 Here and throughout, X ≫ Y means that there exists C > 0 such that X ≥ CY .[15] L. A. Vinh, The Erdős-Facolner d… Show more

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Cited by 3 publications
(3 citation statements)
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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%
“…(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%
“…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%
“…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%