The sum-product problem asks how large the set of sums and products of an arbitrary set of real numbers must grow. I will discuss some recent progress on the sum-product problem for sets of real numbers, in particular explaining how recent insights into bounds for the third moment of convolutions have led to new bounds for this problem.

We study the number of $3$-term arithmetic progressions ($3$-APs) in finite sets $A=\{a_1 <\cdots < a_n\}\subset \mathbb{R}$ of real numbers. The maximum is achieved when $A$ itself forms an $n$-term arithmetic progression, in which case the number of $3$-APs is of order $\Theta(n^2)$. If instead we consider convex sets, i.e. the consecutive differences $d_i=a_{i+1}-a_i$ form a strictly increasing sequence, it seems natural to suspect that in this case the number of $3$-APs is significantly smaller. Indeed, their number is bounded by $O\left(n^{\frac{5}{3}}\right)$. However, we will also present a construction having $\Omega\left(n^{\frac{3}{2}}\right)$ many $3$-APs. We will further discuss connections to Jarník theorem on integral points on convex curves. This is joint work with Thomas F. Bloom and Oliver Roche-Newton.

Developing upon the squeezing method of Ruzsa, Shakan, Solymosi and Szemerédi, it is proved by Hanson, Roche-Newton and Rudnev that given any finite set of real numbers $A$ and any $k$-convex function f (that is any function whose first $k-1$ derivatives are strictly monotone), that at least one of the sets $2^kf(A)-(2^k-1)f(A)$ and $A+A-A$ must be large. In this talk, we extend upon these ideas to prove, for any finite set $A \subset [0,1]$ and any family $\mathcal{F}$ of $n$ functions that are analytic on an open interval containing $[0,1]$ and whose derivatives are linearly independent, the set $2^{n-1}f(A)-(2^{n-1}-1)f(A)$ must be large for at least one $f \in \mathcal{F}$. As a corollary, we obtain improved bounds on the additive growth of the set $f(A-A)$ for a general class of functions and, by augmenting an approach of Roche-Newton, use this to bound the additive growth of the set of angles determined by a Cartesian product set $A \times A$.

It is often useful to cover a set by translates of a “better understood” set. In this talk some known results will be surveyed and some open questions will be discussed.

Given an integer $d>1$ and a finite, non-empty set $X$ of $d \times d$ matrices with integer entries, one is interested in counting the number of matrices $A, B \in X$ such that $AB = BA$. Recently, Browning–Sawin–Wang investigated this question in the case when $d \geq 3$ and $X$ is the set of all integer matrices with entries in $\{-N,...,N\}$. Inspired by their work, I analysed the $d=2$ case of this problem but having replaced $\{-N,...,N\}$ with an arbitrary, finite set of integers. The latter connects nicely to topics in incidence geometry, sum-product phenomenon and growth in groups. In this talk, I will give a survey of the above results and the methods involved, and mention some more progress on this type of question when $d=3$. Part of this involves joint work with Jonathan Chapman.

In his study of the polynomial Hirsch conjecture, Santos introduced the problem of determining the largest possible diameter of a strongly connected d-dimensional simplicial complex on n vertices. We give a precise answer for any fixed d and sufficiently large n. Joint work with Olaf Parczyk, Silas Rathke and Tibor Szabó.

We are interested to lower bound the number of distinct dot products determined by a finite set of points in the Euclidean plane. The best known bound in this direction was achieved by work of B. Hanson, S. Senger and O. Roche-Newton (2023). We use their ideas while employing efficiently the Pl\"{u}nnecke-Ruzsa and Ruzsa triangle inequalities to obtain an improvement on the exponent.

An innocent question of Erdős and Turán (1936) asked about the maximal size of sets in $\{1,2,..., N\}$ without $3$ integers in arithmetic progression. The upper bound on the size of such sets, when $N$ is large, has seen influential ideas, and the problem became one of the corner stones of combinatorial number theory/additive combinatorics. (The list of contributors includes Roth, Szemer\'{e}di, Furstenberg, Bourgain, Gowers, Sanders, Bloom and Sisask, and most recently Kelley and Meka.) On lower bounds the situation remained unclear: Szekeres (1936) conjectured (based on small examples) that the set of integers avoiding the digit 2 in the ternary system might be best possible. Salem and Spencer (1942) and Behrend (1946) gave better constructions. We will explain these previous constructions and give the first significant improvement since about 80 years, in $({\mathbb{Z}}_m)^n$ and in the integers. This is joint work with Zach Hunter, Laura Proske and Lisa Sauermann.