4 problems

Minimizer conjecture. For every $1\leq i\leq n-1$, $\mathbf y_n(i,i+1)$ is the unique element of $\mathcal F_n^+$ such that

Unboundedness conjecture. For every $N\in\mathbb N$, there exists an algebraic number having at least $N$ exceptional points.

Rational attainment conjecture. If $\alpha$ is rational and $t\in(0,\infty]$, then there exist rational points $\alpha_1,\ldots,\alpha_N$ such that

Let $\alpha\in\overline{\mathbb Q}^{\times}$, and define its metric Mahler measure by … Here $M$ denotes the usual Mahler measure, and the infimum ranges over all finite product re…