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

Iosevich, Alex; Koh, Doowon; Parshall, Hans

Mosc. J. Comb. Number Theory 8, No. 2, 103-115 (2019).

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 \(\mathbb F_q^+ =\{x^2: x\in \mathbb F_q,\ x\neq 0\}\).

Reviewer: Noureddine Daili (Sétif)

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

Citations:

Zbl 1270.11001; Zbl 0060.34805; Zbl 1252.52013; Zbl 0864.11063; Zbl 1145.11306

References:

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