On a conjecture by Eriksson concerning overlap in strings

Consider a finite alphabet Omega and strings consisting of elements from Omega. For a given string Mi, let cor(w) denote the autocorrelation, which can be seen as a measure of the amount of overlap in w. Furthermore, let a(w)(n) be the number of strings of length n that do not contain w as a substring. Eriksson [4] stated the following conjecture: if cor(w) > cor(w'), then a(w)(n) > a(w')(n) from the first n where equality no longer holds. We prove that this is true if \Omega\ greater than or equal to 3, by giving a lower bound for a(w)(n) - a(w')(n).