2 problems

Let $d\geq 2$ and $k\geq 1$. Let $A_i$ be positive integers, let $u_i,v_i$ be integers, let $\lambda\ne 0$ be a real constant, and let $L(M)$ be an integer-valued linear polynomial…

Let $d\geq 2$, $k\geq 1$, and $\varepsilon>0$. For $N\geq 1$ and $\beta\in\mathbb R/\mathbb Z$, consider the distance to the nearest integer, denoted by $\|\cdot\|$. Uniform expone…