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 be a cyclotomic integer, and suppose that every conjugate of has absolute value at most . Then what can one say about ? If , then Kronecker’s theorem says that is a root of unity (this statement only requires that is an algebraic integer). Robinson studied the problem of what happens when and also . 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 is an algebraic integer the largest of whose absolute values is , then what are the possible values of ? Two such families of such numbers are those of the form
for a root of unity . These give all of the form
Note that these sets have limit points at and respectively. It turns out that there exactly two further exceptions, as follows:
The first element is totally real and cyclotomic, and so manifestly occurs as such an . The second turns out to be the absolute value of . The proof by Robinson and Wurtz actually applies to slightly larger values of , and after the limit point there is another gap, and the next smallest possible is
The first two exceptional numbers turn up in relation to subfactors. How about the last example?