First, we examine the random process of the form Xn+1 = AXn + Bn (mod p) where A is a fixed 2x2 matrix with entries 2, 1, 1, 1, B_0, B_1, B_2,... are independent and identically distributed on the vectors [0 0], [0 1], [1 0], and X_0 is the 0 vector, with the goal of bounding the rate of convergence of this process to the uniform distribution.

To show the bound of order log p log(log p) (i.e. for n of this size, the variation distance of P_n from uniform is less than epsilon for sufficiently large p), we examine the Fourier transform and its role in this setting. The emergence of the Fibonacci sequence in this random process leads us to introduce an expansion we call the Fibonary expansion useful in analyzing the Fourier transform.

Next, we expand our results to Xn+1 = AXn + Bn (mod p) where A is a fixed non-trivial diagonalizable 2x2 matrix satisfying det(A) = 1. The entries of A are nonnegative integers and A has no eigenvalues of 1 over C . B_0, B_1, B_2,... and X_0 are defined the same as above. Here we introduce what we call the beta-ary expansion to analyze the Fourier transform.