Tropical approximation of exponential sums and the multivariate Fujiwara bound

We prove a multivariate analogue of the Fujiwara bound: for a $d$-variate exponential sum $f$ with exponents having spacing $\mu$, the distance from a point $x$ in the amoeba $\mathscr{A}_f$ to the Archimedean tropical variety of $f$ is at most $d \sqrt{d} 2\log(2 + \sqrt{3})/ \mu$. If all exponents are integral, then the bound can be improved to $d \log(2 + \sqrt{3})$. Both bounds are within a constant factor of optimal.