Arithmetic Gevrey series and the nearest integer to $\log{n}$

Abstract: I will report on the continued joint project with Calegari and Tang on arithmetic holonomy bounds, this time in the broader context of Gevrey series.

We give an extension of Andre's algebraicity criterion to the arithmetic Gevrey class, which turns out to contain the Hermite-Lindemann theorem directly as a special case. Another application solves the following problem first raised by Mahler: We prove that for all integers $n > 1$, the nearest integer distance to $\log{n}$ is bounded below by some universal polynomial in $1/n$.

In the language of the theory of linear forms in logarithms, this supplies precisely the Waldschmidt parameter feature (which has only been proved for homogeneous linear forms with rational coefficients) in the inhomogeneous case of linear forms in logarithms on $\mathbb{G}_a \times \mathbb{G}_m$.

number theory

Audience: researchers in the topic

BC-MIT number theory seminar