Intervals Containing Infinitely Many Sets of Conjugate Algebraic Units

Abstract

In [6], the author proved that all intervals of length 4 are critical for the problem of conjugate algebraic integers. That is, any longer interval contains infinitely many sets of conjugate algebraic integers, whereas any shorter interval contains only a finite number of such sets. For definiteness, we shall consider the critical intervals as being closed intervals. Only the positive result was new, the negative result being classical. It was first proved by Polya, and appeared in Schur [7, p. 391]. A general theory including this result was constructed by Fekete [1]. Some positive results were found by Fekete and Szegd [2], but they do not apply to real point sets, only to two dimensional point sets in the complex plane. The problem remains open for the critical intervals, except when the end points are rational integers. In this case, the interval does contain infinitely many sets of conjugates. For example, the interval [-2, 2] contains the numbers x = 2 cos 2kwr/m with 0 < k < m/2 and (k, m) = 1, which form a set of conjugates for each positive integer m. That is, the polynomial