Gaps in binary cyclotomic polynomials

Abstract

For odd prime numbers $p<q$, let $\hat{I}{¦}_{pq}\in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\hat{I}{¦}_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $\hat{I}{¦}_{pq}$ are described as concatenations of words arising from iterations of a circular map.

This version includes a restructuring of the presentation. The results and proofs remain exactly the same. This paper was accepted for publication in the International Journal of Number Theory