Small Cyclotomic Integers

Julia Robinson is a famous mathematician responsible for fundamental work in logic and in particular on Hilbert’s Tenth problem. Less well known nowadays is that her husband, Raphael Robinson, was a number theorist at Berkeley. One question R.Robinson asked concerned small cyclotomic integers. Namely, let \(\alpha\) be a cyclotomic integer, and suppose that every conjugate of \(\alpha\) has absolute value at most \(R\). Then what can one say about \(\alpha\)? If \(R \le 1\), then Kronecker’s theorem says that \(\alpha\) is a root of unity (this statement only requires that \(\alpha\) is an algebraic integer). Robinson studied the problem of what happens when \(R \le 2\) and also \(R \le \sqrt{5}\). He made five conjectures concerning these questions, four of which were solved in the 60’s by Jones, Cassels, and Schinzel. Five decades later, Frederick Robinson (no relation!) and Michael Wurtz proved the last of these conjectures (while working with me as summer students), and their paper has just been accepted by Acta Arithmetica. In particular, they answer the following problem: if \(\alpha\) is an algebraic integer the largest of whose absolute values is \(R \le \sqrt{5}\), then what are the possible values of \(R\)? Two such families of such numbers are those of the form

\(\zeta + \zeta^{-1}, \qquad i + \zeta + \zeta^{-1}\)

for a root of unity \(\zeta\). These give all \(R\) of the form

\(2 \cos(\pi/N), \qquad \sqrt{1 + 4 \cos^2(\pi/N)}.\)

Note that these sets have limit points at \(\sqrt{4}\) and \(\sqrt{5}\) respectively. It turns out that there exactly two further exceptions, as follows:

\(\displaystyle{\frac{\sqrt{3} + \sqrt{7}}{2}, \qquad \sqrt{\frac{5 + \sqrt{13}}{2}}}\)

The first element is totally real and cyclotomic, and so manifestly occurs as such an \(R\). The second turns out to be the absolute value of \(1 + \zeta_{13} + \zeta^4_{13}\). The proof by Robinson and Wurtz actually applies to slightly larger values of \(R\), and after the limit point \(\sqrt{5}\) there is another gap, and the next smallest possible \(R\) is

\(|1 + \zeta_{70} + \zeta^{10}_{70} + \zeta^{29}_{70}| \sim \sqrt{5.017655 \ldots}\)

The first two exceptional numbers turn up in relation to subfactors. How about the last example?