We consider two problems of Erdo"s, both from Number Theory:
(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.
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.
(Includes joint work with I.Z.Ruzsa and M.B.Nathanson)
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