A set A subset of N is called sparse if, for some No, the distance between any two elements of A is at least N-0. James Currie (2008) [2] showed that for each sparse set A and every subset P subset of A there exists a ternary square-free word w(P) such that a palindrome of length three starts at position i is an element of A in w(P) if and only if i is an element of P. We provide a simpler proof of this result that also works for shorter distances between the positions.