Gaps in binary cyclotomic polynomials

For odd prime numbers [Formula: see text], let [Formula: see text] be the binary cyclotomic polynomial of order [Formula: see text]. In this paper, we prove that the second gap of [Formula: see text] is the maximum of [Formula: see text] and [Formula: see text], where [Formula: see text] is the remainder of [Formula: see text] divided by [Formula: see text]. For [Formula: see text] congruent to [Formula: see text] modulo [Formula: see text], we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of [Formula: see text] are described as concatenations of words arising from iterations of a circular map.