Intersection distribution, non-hitting index and Kakeya sets in affine planes

Abstract

In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size q + 1 in the classical projective plane P G(2, q), where the intersection distribution of S indicates the intersection pattern between S and the lines in P G(2, q). The second one relates to a polynomial f over a finite field Fq, where the intersection distribution of f records an overall distribution property of a collection of polynomials {f (x) + cx | c ∈ Fq}. These two perspectives are closely related, in the sense that each polynomial produces a (q +1)-set in a canonical way and conversely, each (q +1)-set with certain property has a polynomial representation. Indeed, the intersection distribution provides a new angle to distinguish polynomials over finite fields, based on the geometric property of the corresponding (q + 1)-sets. Among the intersection distribution, we identify a particularly interesting quantity named non-hitting index.For a point set S, its non-hitting index counts the number of lines in P G(2, q) which do not hit S. For a polynomial f over a finite field Fq, its non-hitting index gives the summation of the sizes of q value sets {f (x) + cx | x ∈ Fq}, where c ∈ Fq. We derive lower and upper bounds on the non-hitting index and show that the non-hitting index contains much information about the corresponding set and the polynomial. More precisely, using a geometric approach, we show that the non-hitting index is sufficient to characterize the corresponding point set and the polynomial, when it is very close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the non-hitting index and the intersection distribution of several families of point sets and polynomials. As an application, we consider the determination of the sizes of Kakeya sets in affine planes. The polynomial viewpoint of intersection distributions enable us to compute the size of a few families of Kakeya sets with nice algebraic properties.Finally, we describe the connection between these new concepts and various known results developed in different contexts and propose some future research problems.

On the intersection distribution of degree three polynomials and related topics

“…In [13], an exhaustive search determines all possible sizes of Kakeya sets in P G(2, q) where q ≤ 9. Inspired by this work, the authors of [22] proposed explicitly constructions of Kakeya sets with nice underlying algebraic structures, which are derived from monomials over finite fields and have previously unknown sizes. Along this line, we present infinite families of Kakeya sets from degree three polynomials in this section.…”

“…Moreover, as a special case, the multiplicity distribution of x 3 follows from the result of Bluher [3, Theorem 5.6], see also [22,Proposition B.9].…”

“…So far, not much is known about the non-hitting index of monomials. Employing Theorems 1.6 and 2.8, in Table 2.1, we give an update of [22,Table 3.2], where an entry with superscript represents the non-hitting index derived from Theorems 1.6 and 2.8, an entry with superscript ⋆ represents the non-hitting index which has not yet been understood, an entry without superscript represents the non-hitting index known before. Note that in the table, when (d, q − 1) = 1, we group d and its inverse modulo q − 1 together.…”

“…(1) When p = 2, Family (1a) in Theorem 3.1 contains quadratic monomials, whose intersection distribution follows from [3, Theorem 5.6] (see also [22,Proposition B.9]). The monomials in (1a) are permutations whenever m is odd.…”

“…In contrast, the p = 3 case is more interesting since some less obvious monomials occur. On one hand, the Family (2a) contains linearized monomials, whose proof is easy (see for instance [22,Table 3.1]). Moreover, each monomial in Family (2a) has an inverse belonging to the same family.…”