6 problems

Let $S\subseteq\mathbb{Z}$ be an $\operatorname{(\mathcal{A},\mathcal{B})}\text{-expanding}$ set, meaning that it has the expanding property defined in the paper for the classes…

Conjectured improvement. In Theorem, the factor roughly $C\log^4(2/\alpha)$ can be replaced with $C\log(2/\alpha)$.

Green–Tao's conjecture. The set $2A-2A$ contains a Bohr set of rank $C\log(1/\alpha)$ and radius $\alpha^C$.

For $r\in\mathbb N$, let $BH:=BH(\bm\beta,\bm y,k,\varepsilon)$ be a proper Bohr-Hamming Ball, with $\varepsilon>0$ and $y\in\mathbb T^r$. Define … A set is $(\delta,k)$-recurrent…

Let $\alpha_1,\dots,\alpha_k\in\mathbb{R}$, let $h\in\mathbb{Z}[x]$ be an intersective polynomial of degree $\ell$, and let $N\in\mathbb{N}$. Theorem B gives an integer…

Bohr-covering conjecture. There exist such $\mathcal{B}$ and translates $t_1,\ldots,t_k$ for which the symmetric difference between $S$ and the lower level-set has size at most…