On the quotient set of the distance set.(English)Zbl 1454.11222

In this paper the authors review the known Erdös-Falconer distance problem over finite fields [P. Charpin (ed.) et al., Finite fields and their applications. Character sums and polynomials. Based on the invited talks of the RICAM-workshop, Strobl, Austria, September 2–7, 2012. Berlin: de Gruyter (2013; Zbl 1270.11001); P. Erdős, Am. Math. Mon. 53, 248–250 (1946; Zbl 0060.34805); D. Koh and C.-Y. Shen, J. Number Theory 132, No. 11, 2455–2473 (2012; Zbl 1252.52013); R. Lidl and H. Niederreiter, in: Handbook of algebra. Volume 1. Amsterdam: North-Holland. 321–363 (1996; Zbl 0864.11063)]. Namely, Let $$E$$ be a set in $$\mathbb F_d^q$$, and let $$\Delta(E)$$ be the set of distinct distances determined by the pairs of points in $$E$$. How large does $$E$$ need to be to guarantee that $$|\Delta (E)|\gg q$$? The finite field variant of the Erdős distinct distances problem was first studied by J. Bourgain et al. [Geom. Funct. Anal. 14, No. 1, 27–57 (2004; Zbl 1145.11306)] and [P. Charpin (ed.) et al. (loc. cit.)]:
Theorem (Bourgain-Katz-Tao): Suppose $$q\equiv 3\bmod 4$$ is a prime. Let $$E$$ be a set in $$\mathbb F_q^2. If \(|E|=q^{\alpha}$$ with $$0<\alpha<2$$, then we have $$|\Delta(E)|\gg |E|^{(1/2)+\varepsilon}$$, for some positive $$\varepsilon=\varepsilon(\alpha)>0$$. In this work, the authors prove that the exponent $$d/2$$ holds for the quotient set of the distance set. Namely, authors prove the following two theorems in two results for the cases of $$d$$ even and odd integer, in several lemmas, using the nontrivial principal character on $$\mathbb F_q$$, Fourier inversion and Plancherel formula and others from [\textit{R. Lidl} and \textit{H. Niederreiter}, in: Handbook of algebra. Volume 1. Amsterdam: North-Holland. 321--363 (1996; Zbl 0864.11063)]: Theorem 1. Let $$\mathbb F_q$$ be a finite field of order $$q$$, and let $$E$$ be a set in $$\mathbb F_q^d$$. Let $$E\subset \mathbb F_q^d$$, $$d$$ even. If $$|E|\geq 9q^{d/2}$$, then we have $$((\Delta(E))/(\Delta(E)))=\mathbb F_q$$ where $((\Delta(E))/(\Delta(E))) := \{(a/b): a\in\Delta(E), b\in\Delta(E)\{0\}\}$ and $\Delta(E):=\{||x-y||:x,y\in E\},\quad ||x||:=\Sigma x_{i}^2.$ Theorem 2. Let $$d\geq 3$$ be an odd integer and $$E\subset \mathbb F_q^d$$. Then if $$|E|\geq 6q^{d/2}$$, we have $\{0\}\cup \mathbb F_q^+\subset((\Delta(E))/(\Delta(E))),$ where $$\mathbb F_q^+ =\{x^2: x\in \mathbb F_q,\ x\neq 0\}$$.

MSC:

 11T24 Other character sums and Gauss sums 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 11H31 Lattice packing and covering (number-theoretic aspects)

Keywords:

quotient set; distance set; finite field
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References:

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