3 research outputs found
A Szemeredi-Trotter type theorem in
We show that points and two-dimensional algebraic surfaces in
can have at most
incidences, provided that the
algebraic surfaces behave like pseudoflats with degrees of freedom, and
that . As a special case, we obtain a
Szemer\'edi-Trotter type theorem for 2--planes in , provided
and the planes intersect transversely. As a further special case, we
obtain a Szemer\'edi-Trotter type theorem for complex lines in
with no restrictions on and (this theorem was originally proved by
T\'oth using a different method). As a third special case, we obtain a
Szemer\'edi-Trotter type theorem for complex unit circles in . We
obtain our results by combining several tools, including a two-level analogue
of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational
Geometr
A survey on additive and multiplicative decompositions of sumsets and of shifted sets
In this paper we survey results on sumsets with multiplicative prop-erties and the question if a shifted copy of a multiplicatively defined set can again be multiplicatively defined. The methods involved are of analytic nature such as the large sieve, and of combinatorial nature such as extremal graph theory.