Note on a Class of Subsets of AG(3, q) with Intersection Numbers 1, q and n with respect to the Planes

m i are called intersection numbers of K. As usual, by a k-set we mean a set of size k. In the literature one can find many papers devoted to the study of k-sets of given type, not only in affine and projective geometries (cf., e.g., [1–15]), and most of these results are characterizations of classical geometric objects. Recently, characterizations of Hermitian varieties and quadrics of PG(r, q) as k-sets with given intersection numbers with respect to more than one family of subspaces (e.g., with respect to planes and solids) have been considered [16, 17]. A cap of an affine or projective space of dimension ≥3 is a subset of points no three of which are collinear. In 1995,O. Ferri and S. Ferri [18] gave a characterization of


Introduction
Let G denote either a finite projective space or a finite affine space, and let { , . . . , } be a set of nonnegative integers with 0 < 1 < ⋅ ⋅ ⋅ < . A subset K of G is of class [ , . . . , ] with respect to the subspaces of dimension of G if any subspaces intersect K either in , . . . , −1 or points, and K is of type ( , . . . , ) with respect to the subspaces of dimension if for every integer , ∈ {0, . . . , } there exists a -subspace meeting K in exactly points. The numbers are called intersection numbers of K. As usual, by a -set we mean a set of size .
A cap of an affine or projective space of dimension ≥3 is a subset of points no three of which are collinear.
In 1995, O. Ferri and S. Ferri [18] gave a characterization of 2 -caps of AG (3, ) in terms of sets with three given intersection numbers with respect to the planes. Their result reads as follows.
Unfortunately, a step of the proof of that theorem is not correct; in fact it contains a counting argument which does not give the contradiction they want (see [18] page 71 line +7). However, the statement of the result is true as we are going to prove in Lemma 4.
In this paper we will prove the following slight extension of the O. Ferri and S. Ferri result. Thus, it follows that the sets of type (1, , +1) 2 of AG (3, ) have size at most 2 and that equality holds if and only if they are caps.

Proof of Theorem 1
In this section, first, we briefly recall the basic equations for a -set of AG(3, ) with three intersection numbers, and then we will assume that = 2 and we will give the proof of Theorem 1. (1) From (1) it follows that

2 -Sets of AG(3, ) with Three Intersection Numbers.
From now on, K is a -set of AG(3, ) with ≥ 2 and with intersection numbers 1, , and with respect to the planes. Since ≥ 2, there are at least two distinct points in . Let and be two points of , and let ℓ be the line connecting them. Put := |ℓ ∩ |. Namely, ≥ 2.
The proof of Theorem 1 follows from Lemmas 3 and 4. Let us end with the following easy consequence of Theorem 2.