English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We propose the concepts of intersection distribution and non-hitting index,which can be viewed from two related perspectives. The first one concerns apoint set $S$ of size $q+1$ in the classical projective plane $PG(2,q)$, wherethe intersection distribution of $S$ indicates the intersection pattern between$S$ and the lines in $PG(2,q)$. The second one relates to a polynomial $f$ overa 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 closelyrelated, in the sense that each polynomial produces a $(q+1)$-set in acanonical way and conversely, each $(q+1)$-set with certain property has apolynomial representation. Indeed, the intersection distribution provides a newangle to distinguish polynomials over finite fields, based on the geometricproperty of the corresponding $(q+1)$-sets. Among the intersectiondistribution, we identify a particularly interesting quantity named non-hittingindex. 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 indexcontains much information about the corresponding set and the polynomial. Moreprecisely, using a geometric approach, we show that the non-hitting index issufficient to characterize the corresponding point set and the polynomial whenit is close to the lower and upper bounds. Moreover, we employ an algebraicapproach to derive the intersection distribution of several families of pointsets and polynomials, and compute the sizes of related Kakeya sets in affineplanes.

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