Appears in collection : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

A cap is a point set in affine or projective space without any three points on any line. We will discuss the current state of the art, and give an exponential improvement for the size of caps of AG(n, p), which one can think of as (Z/pZ)^n, and PG(n,p). For certain primes, 5,11,17,23,29 and 41, we improve the asymptotic growth of these caps, for example, when p=23 from (8.091...)^n to (9-o(1))^n, as n tends to infinity.