This post is a follow-up on a previous post on the abelian house.
For an algebraic integer \(\alpha\), the house \(\overline{|\alpha|}\) is the absolute value of the largest conjugate of \(\alpha.\) Raphael Robinson made a number of conjectures concerning cyclotomic integers with small house, some of which were solved a long time ago by Jones, Cassels, and Schinzel, and one much more recently by Frederick Robinson (no relation!) and Michael Wurtz as part of an REU, discussed in this post. My description there was actually somewhat inaccurate; the work of Robinson-Wurtz did not fully resolve all of Robinson’s conjecture and indeed one remained outstanding. In one or two email conversations I had with Kiran Kedlaya, it certainly seemed plausible that these methods could be pushed forward to resolve the remaining conjecture, though some work was certainly going to be required. As always,the issue is passing from “effective in principle” to “effective in practice”. I’m happy to say that this conjecture has now been resolved! In a recent paper of Jitendra Bajpai, Srijan Das, Kiran Kedlaya, Nam Le, Meghan Lee, Antoine Leudière, and Jorge Mello, the authors build and improve upon the previous methods to prove Robinson’s conjecture and more. (Added: this was “recent” when the first draft of this blog post was written in 2025 which seems infinitely long ago; and in fact just yesterday Kiran sent me the sequel!) Many of the arguments are similar to those of previous papers, but in this type of problem a small improvement can make the difference between a feasible and an intractable problem. One problem they raise as a future project is to compute the first limit point of houses of algebraic integers that is not itself a house of an algebraic integer, a topic that was the main point of this post. They remark in passing that they found a significantly smaller limit point than the one I had found, namely
\[\frac{25}{4} = 6.25 < \alpha = \displaystyle{\frac{97 + 26 \sqrt{13}}{27} = 7.064604\ldots}\] With some trepidation I checked to see how confidently I had predicted that \(\alpha\) was the smallest such point. Here is the relevant excerpt:
Moreover, I think it quite likely (and quite provable, perhaps with a certain amount of computational effort) that [\(\alpha\)] is the smallest number in \(\overline{{{\mathfrak{M}}^{{\mathrm{ab}}}_{\infty}}} \setminus {{\mathfrak{M}}^{{\mathrm{ab}}}_{\infty}}\).
Not too bad! I didn’t formally conjecture it, or (worse) claim it to be true, so I consider that a success.