### Effective Aspects in Diophantine Approximation

Organizers: B. Adamczewski, A. Bostan, B. Salvy, W. Zudilin

March 27 to 31, 2023

Classifying objects, determining their algebraic relations over a field, are often the culminating point of a mature theory. When the objects are numbers, the topic becomes understandable to a vast majority of people not necessarily from mathematics. At the end, a surprise is an actual difficulty of proving that relations one cannot find do not exist. This is a point where the theories within number theory require an external assistance, namely, effective algorithms to perform certain specific jobs that are out of the human capacity as well as to systematize but also effectivize and automatize the existing techniques and knowledge in a most efficient manner. This strategy is proved to be working well in solving classification problems related to special functions and their values. It is traditional in Diophantine approximation to address arithmetic properties (irrationality, transcendence, linear and algebraic independence over number fields) of numbers like $\pi$, $e$, the values of the Riemann zeta function $\zeta(s)$ and of many other functions that show their appearance throughout mathematics. Of particular interest are, for example, the values of Siegel’s $E$- and $G$-functions, which come as solutions of linear differential equations, as well of Mahler’s functions, for which the differentiation $d/dx$ is replaced by the operator $x\mapsto x^p$. The setup for arithmetic study of the values of such functions is rather classical, however there is a clear demand for investigating underlying differential and functional equations using effective algorithms. The interest of computer algebra in this subject is not recent. For example, Apéry’s celebrated proof of the irrationality of $\zeta(3)$ motivated research on the automatic manipulation with binomial sums, and led to a whole algorithmic paradigm, called creative telescoping. Recently, classical results on linear (in-)dependence of values of $E$-functions at algebraic points have been made effective. Further work on actual efficient algorithms and implementations will open the way to the automatic production of new explicit identities, or automatic proofs of algebraic independence. More generally, the potential of this interaction between computer algebra and Diophantine approximation is still largely unexploited.

This one-week long workshop will bring together experts from the two different communities––computer algebra and number theory––to take stock of the current state-of-the-art, exchange the necessities of two areas, discuss open problems and envisage new directions and joint projects for the future. The idea is to stimulate an emerging trend, whose target is to push the established methods in Diophantine approximation to their systematic algorithmization, in order to apply them well beyond what can be calculated by hand.