3 problems

Let $K_a$ be the simplest cubic field considered in the paper, let $\mathcal{O}_{K_a}$ be its maximal order, and let $u_1,u_2\in\mathcal{O}_{K_a}^*$ and $n\in\mathbb{Z}$ satisfy th…

Let $K_a$ be the simplest cubic field considered in the paper, let $\rho$ generate the order $\mathbb{Z}[\rho]$, and let $u_1,u_2\in\mathbb{Z}[\rho]^*$ and $n\in\mathbb{Z}$. Two so…

Let $A$ be a finite-dimensional division algebra over $\mathbb{Q}$, and let $\Gamma_1,\Gamma_2$ be finitely generated semigroups of the multiplicative group $A^\times$. Fix…