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Pairs of Dot Products in Finite Fields and Rings

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given \(E\subset \mathbb F_q^d\) or \(\mathbb Z_q^d\), we provide bounds on the size of the set

$$\begin{aligned} \left\{ (u,v,w)\in E \times E \times E : u\cdot v = \alpha , u \cdot w = \beta \right\} \end{aligned}$$

for units \(\alpha \) and \(\beta \).

Keywords

  • Dot-product sets
  • Sum-product problem
  • Finite fields

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  • DOI: 10.1007/978-3-319-68032-3_8
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Notes

  1. 1.

    This is just in the case that (1, 1) is on one of the lines or \(\alpha =\beta \).

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Covert, D., Senger, S. (2017). Pairs of Dot Products in Finite Fields and Rings. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_8

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