Leírás
Abstract:
We give a survey of recent work especially on lower bounds and some
exact values on the size of sets without any arithmetic progression of
length k
in (Z/mZ)^n, improving for example a lower bound, when k=3, m=4 from
2.519^n to 3^n/sqrt(n).
The case k=3, m=3 corresponds to caps.
A cap is a point set in affine or projective space without any three
points on any line. We discuss the current state of the art, and give
an exponential improvement for the size of caps of in the affine space
AG(n, p),
which one can think of as (Z/pZ)^n, and in the projective space PG(n,p).
Lower bounds on cap constructions go back to Bose (1947), of size
p^(2n/3), and Bierbrauer and Edel (2004), of size about
(p^4+p^2-1)^(n/6).
For the primes, 5,11,17,23,29 and 41, we improve the asymptotic growth
of these caps, for example, when p=23:
Bose's work implies (8.087...)^n, Bierbrauer and Edel's work gives
(8.091...)^n. Our new lower bound is (9-o(1))^n, as n tends to
infinity.
The upper bounds have been of type O(p^n/n) until a few years ago,
when they were improved to O( (c_p p)^n)$,
where c_p < 1 is a constant, by
breakthrough work of Croot, Lev, Pach, and then Ellenberg and Gijswijt.
This is joint work with
P.P.Pach, G.Lipnik, B.Klahn and J. Führer.
The talk should be of interest for the groups in number theoery and
combinatorics.
For Zoom access please contact Andras Biro (biro.andras[a]renyi.hu).