Montreal, December 4 - 7, 2020
A variety of constructions for very dense Sidon sets exists in the additive combinatorics literature, and seemingly follow no shared pattern except that they all "come from algebra".
In this talk I will explain that they fit into a common framework: they all arise from letting $H$ act on a finite projective plane by collineations.
These ideas essentially appeared a long time ago in the design theory literature, but seem less well known in additive combinatorics, so this talk functions as a sort of public service announcement. I will also discuss some related open questions.
Joint work with Sean Eberhard.
Granville--Shakan recently showed that, if $N$ is large enough, the answer to this question is yes, equality does hold. However, for all $d \geqslant 2$ their results gave only an ineffective lower bound on what 'large enough' should mean. In this talk we will describe two new pieces of work on this question: a new bound in the case $d=1$, which is tight for several infinite families of sets $A$, and the first effective bounds for arbitrary $A$ when $d \geqslant 2$. These results are joint work with Granville and with Granville--Shakan respectively. If time permits, we will describe the connections between this work and Khovanskii's theorem (that the size of $NA$ is a polynomial in $N$, for large enough $N$).