7 problems

Znam–Korec conjecture. One has

A system of residue classes is a finite collection $A=\{a_s(n_s)\}_{s=1}^k$ of residue classes of $\mathbb Z$, where $a_s(n_s)=a_s+n_s\mathbb Z$ and the moduli $n_1,\ldots,n_k$ are…

Let $A\subset \mathbb{Z}$ be a finite set of integers and let $m\geqslant 1$. Write $mA=A+\cdots+A$ and $A^{(m)}=A\cdots A$. Iterated sum-product conjecture. One should have … Bour…

Let $\mathcal{Z}$ be a dp-minimal proper expansion of $(\mathbb{Z},+,0,1)$. A classification conjecture for dp-minimal expansions of the integers. Exactly one of the following hold…

Three-translate linear stability conjecture. For every $\delta>0$ there exists $\varepsilon>0$ such that, if

Scaling-invariance conjecture. The same conclusion holds when $B$ is replaced by $\lambda B$ for any integer $\lambda$.

Let $A$ be a $K$-approximate group in $\mathbb{Z}$. A symmetric generalized arithmetic progression is a set of the form … Here $o(1)$ denotes a quantity bounded in magnitude by…