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

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) |

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\textit{A. Iosevich} et al., Mosc. J. Comb. Number Theory 8, No. 2, 103--115 (2019; Zbl 1454.11222)

### References:

[1] | 10.1515/forum-2015-0251 · Zbl 1429.52020 |

[2] | 10.1007/s00039-004-0451-1 · Zbl 1145.11306 |

[3] | 10.1007/s00209-011-0852-4 · Zbl 1269.11011 |

[4] | 10.1090/S0002-9947-2010-05232-8 · Zbl 1244.11013 |

[5] | 10.1515/FORUM.2010.025 · Zbl 1193.42068 |

[6] | 10.1090/S0002-9947-07-04265-1 · Zbl 1145.11083 |

[7] | ; Lidl, Finite fields. Finite fields. Encyclopedia of Mathematics and its Applications, 20, (1997) |

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