Sets with integral distances in finite fields

Abstract

Abstract. Given a positive integer n, a finite field F q of q elements (q odd), and a non-degenerate quadratic form Q on F n q , in this paper we study the largest possible cardinality of subsets E ⊆ F n q with pairwise integral Q-distances; that is, for any two vectors x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ E, one has Q(x − y) = u 2 for some u ∈ F q .

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

“…We fix a non-square element λ ∈ * q , then it is well known that (see, for example, [1,4]) any non-degenerate quadratic form Q on n q can be reduced (by repeated change of variables) to one of the forms Q n,ε , ε ∈ {1, λ}, depending on the value of χ(Q), where for x = (x 1 , . .…”

Integral automorphisms of affine spaces over finite fields

“…Recently, the finite field analog of the classical probem about integral point sets in R n has attracted considerable attention. See, for example, [5] and the references therein. Besides integral point sets, permutations, preserving the integral distances, are also considered in [7,8,9,10].…”

“…Remark 1. We would like to note that in our earlier approach we gave a proof of Theorem 2, which also relies on Lemmas 1-3, but instead of invoking Lester's result (Theorem 3), we used the results of Iosevich et al [5] on maximum point sets with any two of its points being at distance 0. Here we give an outline.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…Furthermore, Schoen & Shkredov [167] have successfully used a "cubic" generalization of the energy. We also have had to leave out such exciting areas of additive combinatorics in finite fields as • the Erdős distance problem [83,94,117,130,132,144,145] as well as its modification in some other settings (distinct volumes, configurations, and so on defined by arbitrary sets in F n q ) and metrics [14,64,142,195,200,202,203,205]; • the Kakeya problem and other related problems about the directions defined by arbitrary sets in vector spaces over a finite field, see [84-86, 88, 128, 131, 151]; • estimating the size of the sets in a finite field that avoid arithmetic or geometric progressions, sum sets and similar linear and non-linear relations; in particular these results include finite field analogues of the Roth and Szemerédi theorems, see [1,6,8,12,77,81,112,113,153,154,158,181]; • estimating the size of the sets in vector spaces over a finite field that define only some restrictive geometric configurations such as integral distances, acute angles, and pairwise orthogonal systems, see [75,133,134,183,194,204]; • distribution of the values of determinants and permanents of matrices with entries from general sets, see [74,196,197]; and several others.…”