Uniqueness of the distribution of zeroes of primitive level sequences over Z/(pe)

Let p be a prime number, p⩾5, Z/(pe) the integer residue ring, e⩾2, Γ={0,1,…,p−1}. For a sequence ā over Z/(pe), there is a unique decomposition ā=ā0+ā1·p+⋯+āe−1·pe−1, where āi be the sequence over Γ. Let f(x) be a primitive polynomial with degree n over Z/(pe), ā and b̄ sequences generated by f(x) over Z/(pe), ā≠0(modpe−1); we prove that the distribution of zeroes in the sequence āe−1=(ae−1(t))t⩾0 contains all information of the original sequence ā, that is, if ae−1(t)=0 if and only if be−1(t)=0 for all t⩾0, then ā=b̄. As a consequence, we have the following results: (i) two different primitive level sequences are linearly independent over Z/(p); (ii) for all positive integer k, āe−1k=b̄e−1k if and only if ā=b̄.