On the number of occurrences of all short factors in almost all words

We previously proved that almost all words of length n over a finite alphabet A with m letters contain as factors all words of length k(n) over A as n→∞, provided limsup_n→∞ k(n)/log n<1/log m.

In this note it is shown that if this condition holds, then the number of occurrences of any word of length k(n) as a factor into almost all words of length n is at least s(n), where lim_n→∞ log s(n)/log n=0. In particular, this number of occurrences is bounded below by C log n as n→∞, for any absolute constant C>0.