Existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field via Arakelov geometry

We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite tuple of arithmetic $\mathbb{R}$-divisors approximate a given tuple of real numbers. We use this result to prove existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field (abbreviated GVF). We introduce GVF functionals on the space of arithmetic $\mathbb{R}$-divisors and interpret the essential infimum function as the infimum of values of normalised GVF functionals, at least when the generic part of the arithmetic $\mathbb{R}$-divisor is big. We also give a new criterion on equality in one of the Zhang's inequalities.