Asymptotic low slopes of toric exponential sums

Let $\mathrm{\Delta }$ be an integral Newton polytope containing the origin. We prove that the generic Newton slopes corresponding to Hodge slopes strictly smaller than one differ from their Hodge values by $O_{\mathrm{\Delta }}(p^{-1})$. No hypothesis is imposed on the semigroup generated by the lattice points of $\mathrm{\Delta }$. The proof combines an auxiliary prime and the Cartier operator to show that a coefficient of each required Dwork principal minor is nonzero. When the origin is interior to $\mathrm{\Delta }$, duality gives the same estimate for the corresponding high slopes and, in dimension two, for all slopes.