Towards a General Theory of Simultaneous Diophantine Approximation of Formal Power Series: Multidimensional Linear Complexity

We model the development of the linear complexity of multisequences by a stochastic infinite state machine, the Battery–Discharge– Model, BDM. The states s ∈ S of the BDM have asymptotic probabilities or mass μ∞(s) = P(q,M) −1 ·q−K(s), whereK(s) ∈ N0 is the class of the state s, and P(q,M) = ∑ K∈N0 PM (K)q −K = ∏M i=1 q i/(qi − 1) is the generating function of the number of partitions into at most M parts. We have (for each timestep modulo M + 1) just PM (K) states of class K. We obtain a closed formula for the asymptotic probability for the linear complexity deviation d(n) := L(n)− ⌈n ·M/(M + 1)⌉ with γ(d) = Θ ( q ) ,∀M ∈ N,∀d ∈ Z. The precise formula is given in the text. It has been verified numerically for M = 1, . . . , 8, and is conjectured to hold for all M ∈ N. From the asymptotic growth (proven for all M ∈ N), we infer the Law of the Logarithm for the linear complexity deviation, − lim inf n→∞ da(n) log n = 1 (M + 1) log q = lim sup n→∞ da(n) log n , which immediately yields La(n) n → M M + 1 with measure one, ∀M ∈ N, a result recently shown already by Niederreiter and Wang.