Higher amalgamation in ${\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}}^{+}$

We show two results on higher amalgamation in the theory ${\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}}^{+}$, the model companion of the theory of difference fields with an additive character (added as a continuous logic predicate) on the fixed field in characteristic 0. On one hand, we show that the non-trivial condition for 3-amalgamation established in a preceding paper is not sufficient for 4-amalgamation. On the other hand, we show that when working over substructures whose ${\mathcal{L}}_{\sigma }$-reduct is a model of $\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}$, $n$-amalgamation holds for all $n\ge 3$.