Asymptotic Properties of the Factors of Words Over a Finite Alphabet

Let A be an alphabet of cardinality m, n k be a sequence of positive integers and ( ) n w A w k ∈ = . In this paper it is shown that if lim supn→ ∞ kn/ ln n < 1/ lnm then almost all words of length n over A contain the factor w, but if lim supn→ ∞ kn/ln n> 1/ lnm then this property is not true. Also, if lim infn→ ∞ kn/ lnn >1/ lnm then almost all words of length n over A do not contain the factor w. Moreover, if lim (ln ln ) n n n k m α → ∞ − = ∈ ¡ then limsup ( , , , / 1 exp( exp( )) n n n W n k w A m α → ∞ ≤ − − and liminf ( , , , / 1 exp( (1 1/ ) exp( )) n n n W n k w A m m α → ∞ ≥ − − − , where W(n, kn,w,A) denotes the set of words of length n over A containing the factor w of length n k .