Let $\mathcal {F}$ denote the set of functions $f \colon [-1/2,1/2] \to \mathbb {R}_{\geq 0}$ such that $\int f = 1$. We determine the value of $\inf _{f \in \mathcal {F}} \| f \ast f \|_2^2$ up to a $4 \cdot 10^{-6}$ error, thereby making progress on a problem asked by Ben Green. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of $B_h[g]$ sets for $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.

We show that there is an absolute $c>0$ such that if $A$ is a finite set of integers, then there is a set $S\subset A$ of size at least $\log ^{1+c}|A|$ such that the restricted sumset $\{s+s^{\prime }:s,s^{\prime }\in S\text{ and }s\neq s^{\prime }\}$ is disjoint from $A$. (The logarithm here is to base $3$.)

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets of an abelian group.

We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$, all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$, are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$.