| Abstract: Let $K$ be a compact set in the complex plane which is invariant under complex conjugation. The famous Fekete--Szeg\H{o} theorem determines whether there exists infinitely many algebraic integers whose Galois orbits are totally contained in (a small neighbourhood of) $K$. When there are indeed infinitely many algebraic integers totally in $K$, recent results by Smith and Orloski--Sardari could characterize all the possible limit distributions. Beyond this setting, one can also work with non compactly supported measures and ask whether they can be approximated by Galois orbits of algebraic integers. This is worked out by Burgos-Gil--Menares--Qu--Sombra, which is indeed just SOS+$\epsilon$. Given all these results, it's fair to say that the problem of "limit distribution of algebraic integers in the complex plane" is more or less well understood. In this talk, I will discuss generalizations of this problem in two directions: we work with arbitrary open curves instead of $\mathbb{A}^1$, and we work adelically at all places instead of only at the archimedean place. We prove that any integral GVF functional can be approximated by integral points in the GVF topology. I will explain what this means, and why this answers the problem of limit distribution of integral points. If time permits, I will present as an application the proof of Montgomery's conjecture on integer Chebyshev constants. This is joint work with Chengyuan Yang (PhD in Peking University). |
Mathematics Seminars and Colloquia