Jumps of ternary cyclotomic coefficients

Abstract

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $\Phi_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to prove that the number of nonzero coefficients of the $n$th ternary cyclotomic polynomial is greater than $n^{1/3}$.