In a previous post, I discussed a new result of Smith which addressed the following question: given a measure \(\mu\) on \(\mathbf{R}\) supported on some finite union of intervals \(\Sigma\), under what conditions do there exist polynomials of arbitrarily large degree whose roots all lie in \(\Sigma\) whose distribution (in the limit) converge to \(\mu\)?
A natural generalization is to replace \(\Sigma\) by a subset of \(\mathbf{C}\) subject to certain natural constrains, including that \(\mu\) is invariant under complex conjugation. I decided that this had a chance of being a good thesis problem and scheduled a meeting with one of my graduate students to discuss it. Our meeting was scheduled at 11AM. Then, around 9:30AM, I read my daily arXiv summary email and noticed the preprint (https://arxiv.org/abs/2302.02872) by Orloski and Sardari solving this exact problem! There are a number of other very natural questions along these lines of course, so this was certainly excellent timing. When I chatted with Naser over email about this, he mentioned he had become interested in this problem (in part) by reading my blog post!
There is of course a general danger of giving my students problems related to my blog posts, and indeed I have refrained from posting a number of times on possible thesis problems, but in this case everything turned out quite well.