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

AbstractWe 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 limsupn→∞k(n)/logn<1/logm.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 limn→∞logs(n)/logn=0. In particular, this number of occurrences is bounded below by Clogn as n→∞, for any absolute constant C>0