Avoiding Forbidden Sequences by Finding Suitable Initial Values

We present a short argument that for a wide class of recurrence relations of the form xn+1 = h(xn, . . . , xn−k) and for an arbitrary fixed (forbidden) sequence (bn) the sequence (xn) avoids (bn) (i.e., ∀n ∈ N xn 6= bn) for almost all initial values (x−k, . . . , x0) ∈ R, excluding a set of Lebesgue measure 0. These sequences include algebraic recurrences and our result also generalizes the affirmative answer to Professor Ladas’ [5, Conjecture 4.5.1]. We also consider some further generalizations on avoiding sequences of countable sets. AMS Subject Classifications: 39A10.