4 problems

Baker's conjecture. The exponent in this bound should be improvable to $1+o(1)$, meaning that one should have

Let $\\{x\\}$ denote the fractional part of a real number $x$. Mahler's conjecture. There does not exist a nonzero real number $\xi$ such that … for all positive integers $n$. This…

Let $n>1$, let $\boldsymbol\alpha\in\mathbb R^n$, and let $\boldsymbol T=(T_1,\dotsc,T_n)\in[1,+\infty)^n$. Write $\bar T=T_1\dotsm T_n$, and let…

Let $\alpha\in\mathbb{R}$, let $k\in\mathbb{N}$, and let $\epsilon>0$. Write $\|\cdot\|$ for the distance to the nearest integer, and let $\sigma$ denote an exponent for which … fo…