Dating back to the 19th century, De Forest’s problem asks: For any finitely supported function on the integer lattice, can we determine the asymptotic behavior of its iterated convolution powers? Following work of T. N. E. Greville, I. J. Schoenberg, V. Thom ́ee, P. Diaconis and L. Saloff-Coste, De Forest’s problem was shown (in joint work with L. Saloff-Coste) to have an affirmative answer in the form of sup-norm asymptotics and local limit theorems. In these local limits, one commonly finds “attractors” given by oscillatory integrals, e.g., the Airy function and the heat kernel evaluated at imaginary time. When this problem is considered on the d-dimensional integer lattice (for d > 1), similar oscillatory integrals appear and influence both sup-norm asymptotics and local limit theorems. In this talk, I will discuss these oscillatory integrals and their appearance in sup-norm estimates and local limits theorems for the convolution powers of complex-valued functions on the d-dimensional integer lattice. Much of this work is joint with Huan Q. Bui.