The asymptotic normalized linear complexity of multisequences

We show that the asymptotic linear complexity of a multisequence a∈FqM∞ that is I≔liminfn→∞La(n)n and S≔limsupn→∞La(n)n satisfies the inequalitiesMM+1⩽S⩽1andM(1-S)⩽I⩽1-SM,if all M sequences have nonzero discrepancy infinitely often, and all pairs (I,S) satisfying these conditions are met by 2ℵ0 multisequences a. This answers an Open Problem by Dai et al.