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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We establish improved finite field Szemeredi-Trotter and Beck type theorems.First we show that if P and L are a set of points and lines respectively in theplane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1N^{3/2-1/662+o(1)} incidences between points in P and lines in L. Here C_1 issome absolute constant greater than 1. This improves on the previouslybest-known bound of C_1 N^{3/2-1/806+o(1)}.  Second we show that if P is a set of points in \mathbb{F}_p^2 with |P|<p theneither at least C_2|P|^{1-o(1)} points in P are contained in a single line, orP determines least C_2 |P|^{1+1/109-o(1)} distinct lines. Here C_2 is anabsolute constant less than 1. This improves on previous results in two ways.Quantitatively, the exponent of 1+1/109-o(1) is stronger than the previouslybest-known exponent of 1+1/267. And qualitatively, the result applies to allsubsets of F_p^2 satisfying the cardinality condition; the previouslybest-known result applies only when P is of the form P=A*A for A \subseteq F_p.

Further improvements to incidence and Beck-type bounds over prime finite fields