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We use the differentiability of the arithmetic volume function and anarithmetic Bertini type theorem to classify when one can find a closed point onthe generic fiber of an arithmetic variety, whose heights with respect to somefinite tuple of arithmetic $\mathbb{R}$-divisors approximate a given tuple ofreal numbers. We use this result to prove existential closedness of$\overline{\mathbb{Q}}$ as a globally valued field (abbreviated GVF). Weintroduce GVF functionals on the space of arithmetic $\mathbb{R}$-divisors andinterpret the essential infimum function as the infimum of values of normalisedGVF 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 ofthe Zhang's inequalities.

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