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

TL;DR

This paper introduces intersection distribution and non-hitting index concepts to analyze point sets and polynomials in finite projective planes, providing bounds and characterizations, and exploring their relation to Kakeya sets in affine planes.

Contribution

It defines the non-hitting index and explores its properties, offering new geometric and algebraic tools to distinguish polynomials and point sets over finite fields.

Findings

Non-hitting index bounds are established.

Characterization of point sets and polynomials near bounds.

Sizes of Kakeya sets in affine planes are computed.

Abstract

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 $PG(2,q)$, where the intersection distribution of $S$ indicates the intersection pattern between $S$ and the lines in $PG(2,q)$. The second one relates to a polynomial $f$ over a finite field $\mathbb{F}_q$, where the intersection distribution of $f$ records an overall distribution property of a collection of polynomials $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. 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 $PG(2,q)$ which do not hit $S$. For a polynomial $f$ over a finite field $\mathbb{F}_q$, its non-hitting index gives the summation of the sizes of $q$ value sets $\{f(x)+cx \mid x \in \mathbb{F}_q\}$, where $c \in \mathbb{F}_q$. We derive 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 close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the intersection distribution of several families of point sets and polynomials, and compute the sizes of related Kakeya sets in affine planes.