SUM/DIFFERENCE/PRODUCT SETS AND GEOMETRY
by Gyorgy Elekes
We consider two problems of Erdo"s, both from Number Theory:
The lower bounds known for either case are only slightly higher
than linear (!). We propose a method using Combinatorial Geometry
which provides much better bounds (with very simple proofs),
together with some generalizations.
- (a) Let there be given N real numbers. Consider the
pairwise sums and also the pairwise products formed
using these numbers. Show that at least one of these
sets is large, i.e. of size close to quadratic,
as a function of N.
- (b) Consider a sequence of N reals where the consecutive
differences strictly increase (the sequence is "convex").
Show that the set of pairwise differences formed using
the N numbers is close to quadratic, as a function of N.
(Includes joint work with I.Z.Ruzsa and M.B.Nathanson)
!!! Dieses Dokument stammt aus dem ETH Web-Archiv
und wird nicht mehr gepflegt !!!
!!! This document is stored in the ETH Web archive
and is no longer maintained !!!