On convexity and sumsets.

Studying additive properties of convex sets is a notable theme in additive combinatorics. A finite sets of reals $A$ is convex when the sequence of differences between its neighbours is strictly monotone, and this presents an obstruction to $A$ being additively structured. In other words, $A$ can be thought as $A=f[N],$ where $f$ has a monotone derivative and $[N]$ is the integer interval.
There are at least two directions to generalise this. One is to replace $f$ by a function with more than one monotone derivative, that is study additive properties of "hyperconvex"sets, assuming that they will have the less additive structure, the more convex $f$ is. The other is to replace $[N]$ by any set of reals $B$ with a small sumset. The latter issue turns out to be quite easy, with the natural measure of additive structure in $B$ being the cardinality ratio $|B-B+B|/|B|. $ As to being able to analyse higher convexity, we can do it (so far) only by considering $2^k$ - fold sums of $A$ with itself, and using induction in $k.$ I'll review some of the results obtained in the last two years along these lines with some sketches of proofs and the emphasis on using elementary order and pigeonhole principle based counting methods, which do not rely on geometric incidence theory. This approach, apart from having produced a reasonably coherent picture within its capability (which is sufficient to at least replicate all the results I am aware of that relied on a version of the Szemerédi-Trotter theorem) has allowed for some small improvements towards one or two milestone open questions. One of those was to do better than $\gg N^{2/3}$ in the question on the minimum number of distinct dot products, defined by pairs of vectors in the $N$-point set in the real plane.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000