### Effective Aspects in Diophantine Approximation

*March 27 to 31, 2023*

*Salle Fokko du Cloux* of the *Bâtiment Braconnier* of the Institut Camille Jordan of the University Lyon 1. Here are directions to the building.

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

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 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.

**Contact:** eada2023@ens-lyon.fr

**Registration: closed since March 3rd**

**Preliminary Program**

Monday 27/03/2023 | ||
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09:30-10:30 | Anthony Poëls |
## Padé approximation for a class of hypergeometric functionsIn a recent work in collaboration with Makoto Kawashima with established new (effective) irrationality measures for values of functions which belong to a certain class of hypergeometric functions (including shifted logarithmic functions and binomial functions). In this talk, we will try to explain the ideas behind our proofs and constructions. |

10:30-11:00 | Welcome coffee+tea | |

11:00-12:00 | Henri Cohen | ## Modular, algebraic, and $\Gamma$-evaluations of hypergeometric seriesWe first give conjecturally complete parametric evaluations of Gauss's $_2F_1$ hypergeometric functions in terms of finite sums of gamma quotients. We then generalize the connection due to Fricke with the Eisenstein series $E_4$ to give functional evaluations of $_2F_1$ corresponding to hyperbolic triangle groups. Finally, we show how a long search has enabled us to find almost $1000$ evaluations of $_2F_1(a,b,c,z)$ with all parameters rational and the result algebraic, and mention that they are probably $90\%$ complete for arithmetic triangle groups. |

12:30-15:00 | lunch | |

15:00-16:00 | Yann Bugeaud | ## Continued fraction expansions of algebraic power series over a finite fieldAlmost nothing is known on the continued fraction expansion of an algebraic real number of degree at least three. The situation is different over the field of power series $\mathbb{F}_p((x^{-1}))$, where $p$ is a prime number. For instance, there are algebraic power series of degree at least three whose sequence of partial quotients have bounded degree. And there are as well algebraic power series of degree at least three which are very well approximable by rational fractions: the analogue of Liouville's theorem is best possible in $\mathbb{F}_p((x^{-1}))$. Recently, in a joint work with Han (built on a previous work by Han and Hu), we proved that, for any distinct nonconstant polynomials $a, b$ in $\mathbb{F}_2 [x]$, the power series $$ [a; b, b, a, b, a, a, b, \ldots ] = a + \frac{1}{b + \frac{1}{b + \cdots}} , $$ whose sequence of partial quotients is given by the Thue–Morse sequence, is algebraic of degree $4$ over $\mathbb{F}_2 (x)$. We discuss this and related results. Furthermore, we give a complete description of the continued fraction expansion of the algebraic power series $(1 + x^{-1})^{j/d}$ in $\mathbb{F}_p((x^{-1}))$, where $j, d$ are coprime integers with $d \ge 3$, $1 \le j < d/2$, and $\gcd(p, jd) = 1$. (Joint work with Han). |

16:00-16:30 | coffee+tea break | |

16:30-17:30 | Vesselin Dimitrov | ## Products of two logarithmsWe explain in detail how to build upon Apéry's method in the framework of arithmetic holonomy bounds to prove the $\mathbb Q$-linear independence of $1, \log(1+1/m), \log(1+1/n)$, and $\log(1+1/m)\log(1+1/n)$ for the range $0 < |1-m/n| < \epsilon_0$ (a positive absolute constant), and how to upgrade the qualitative proof to a quantitative linear independence measure. We further discuss the relations to the classical theory of $G$-functions. This is a report on a joint work with Frank Calegari and Yunqing Tang. |

Tuesday 28/03/2023 | ||

09:30-10:30 | Nicolas Brisebarre | ## Integer points close to a transcendental curve and correctly-rounded evaluation of a functionDespite several significant advances over the last 30 years, guaranteeing the correctly rounded evaluation of elementary functions, such as the cosine, the exponential or the cube root for example, remains a difficult problem. It can be formulated as a Diophantine approximation problem, called the table maker's dilemma, which consists in determining points with integer coordinates close to a curve. In a recent work, we propose two algorithmic approaches to tackle this problem. They are closely linked to a celebrated work by Bombieri and Pila and to the so-called Coppersmith method, which has become classic in cryptography. We will present the initial context, one of our approaches and the results of practical experiments. This is joint work with Guillaume Hanrot (ENS Lyon). |

10:30-11:00 | coffee+tea break | |

11:00-12:00 | Pierre Lairez |
## Numerical periods and effective algebraic geometryBuilding on symbolic integration and numerical evaluation of D-finite functions, we can compute numerically periods of rational integrals with high precision. I will show the main lines of the method and develop three applications: (1) computing the volume of semialgebraic sets; (2) uncovering algebraic curves lying on complex surfaces; (3) computing the singular homology of projective hypersurfaces. This is based on joint work with Marc Mezzarobba, Eric Pichon-Pharabod, Mohab Safey El Din, Emre Sertöz, and Pierre Vanhove. |

12:30-15:00 | lunch | |

15:00-16:00 | Thomas Dreyfus |
## Differential Galois theory and differential transcendence: the exponential caseWe consider meromorphic solutions of difference equations and prove that very few among them satisfy an algebraic differential equation. The basic tool is the difference Galois theory of functional equations. |

16:00-16:30 | coffee+tea break | |

16:30-17:30 | Mark van Hoeij | ## A saga on a generating function of the squares of Legendre polynomialsWe decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy |

17:30-18:30 | Open problem session (see detailed list). | |

Wednesday 29/03/2023 | ||

09:30-10:30 | Masha Vlasenko | ## Frobenius structure and $p$-adic zeta functionI will explain how differential operators coming from algebraic geometry produce interesting $p$-adic numbers. In a recent work with Frits Beukers we give examples of families of Calabi-Yau hypersurfaces in $n$ dimensions, for which one observes $p$-adic zeta values $\zeta_p(k)$ for $1 < k < n$. Appearance of $p$-adic zeta values for differential operators of Calabi-Yau type was conjectured by Candelas, de la Ossa and van Straten. |

10:30-11:00 | coffee+tea break | |

11:00-12:00 | Stéphane Fischler | ## Effective algebraic independence of values of $E$-functionsThe class of $E$-functions has been introduced by Siegel in 1929; it contains the exponential and Bessel functions. Given a finite family of algebraically independent $E$-functions, we consider the set $S$ of algebraic points at which their values are algebraically dependent. The Siegel-Shidlovskii theorem, proved in 1955 and refined by several authors, implies that $S$ is finite. The aim of this talk is to give an algorithm that allows one to determine $S$. It is a joint work with Tanguy Rivoal. |

12:30-15:00 | lunch | |

15:00-20:00 | free afternoon (coffee+tea at 16:00 for those who stay) | |

20:00 | dinner at Le Bistrot Stalingrad | |

Thursday 30/03/2023 | ||

09:30-10:30 | Éric Delaygue | ## A Lindemann-Weierstrass theorem for $E$-functions$E$-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After discussing the recent effective results on the algebraic values of $E$-functions at algebraic points, I will present a generalization of the linear formulation of the Lindemann-Weierstrass theorem for $E$-functions. As a consequence, I will show that the transcendental values at algebraic points of an entire hypergeometric function are linearly independent over the field of algebraic numbers. If time permits, I will give applications to the determination of linear relations between transcendental values of $E$-functions at algebraic points. |

10:30-11:00 | coffee+tea break | |

11:00-12:00 | Tanguy Rivoal | ## Linear independence of values of $G$-functionsIn this talk, I will present recent results obtained with Stéphane Fischler concerning the values taken at algebraic points by families of $G$-functions. These results are dual in some sense to a classical result of Chudnovsky. Our results hold not only for points inside the disk of convergence of a given $G$-function (the usual situation), but also for points in a suitable star-shaped domain at the origin to which the $G$-function can be extended. |

12:00-15:00 | lunch | |

15:00-16:00 | James Worrell | ## Skolem Meets SchanuelThe celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the $p$-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool |

16:00-16:30 | coffee+tea break | |

16:30-17:30 | Christoph Koutschan | ## Tweaking the Beukers integrals in search of more miraculous irrationality proofs à la ApéryThere are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or $\zeta(5)$. Inspired by Frits Beukers' elegant rendition of Apéry's seminal proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$, and heavily using algorithmic proof theory, we systematically searched for other similar integrals, that lead to irrationality proofs. We found quite a few candidates for such proofs, including the square-root of $\pi$ times $\Gamma(7/3)/\Gamma(-1/6)$ and $\Gamma(19/6)/\Gamma(8/3)$ divided by the square-root of $\pi$. Joint work with Robert Dougherty-Bliss and Doron Zeilberger. |

Friday 31/03/2023 | ||

09:30-10:30 | François Brunault | ## Regulator integrals and triple modular valuesRegulator integrals are a certain kind of periods which arise in Beilinson's conjectures on special values of $L$-functions. In the case of modular forms, the regulator integrals introduced by Beilinson involve products of two Eisenstein series. We will describe a new regulator integral based on the Goncharov regulator map. Using triple modular values, we can compute this integral as the $L$-value of a weight $2$ modular form at $s=3$. As an application, we prove a conjecture of Boyd and Rodriguez Villegas on the Mahler measure of $(1+x)(1+y)+z$. This is (in part) joint work with Wadim Zudilin. |

10:30-11:00 | coffee+tea break | |

11:00-12:00 | Francis Brown |
## Geometric and motivic approaches to irrationalityI will explain how ideas from algebraic geometry can inform the construction of rational approximations to zeta values, with an emphasis on the role of Poincaré duality. I will also report on joint work with Wadim Zudilin on rational approximations to $\zeta(5)$. |

12:30-15:00 | lunch |

**Participants**

- Boris Adamczewski
- Gabriele Bogo
- Alin Bostan
- Nicolas Brisebarre
- Francis Brown
- François Brunault
- Yann Bugeaud
- Frédéric Chyzak
- Henri Cohen
- Eric Delaygue
- Vesselin Dimitrov
- Thomas Dreyfus
- Stéphane Fischler
- Florian Fürnsinn
- Colin Faverjon
- Louis Gaillard
- Nutsa Gegelia
- Laurent Habsieger
- Guillaume Hanrot
- Neelam Kandhil
- Jakub Konieczny
- Christoph Koutschan
- Pierre Lairez
- Joris Nieuwveld
- Hadrien Notarantonio
- Armand Noubissie
- Eric Pichon-Pharabod
- Anthony Poëls
- Berend Ringeling
- Tanguy Rivoal
- Julien Roques
- Bidisha Roy
- Ekata Saha
- Bruno Salvy
- Catherine St-Pierre
- Warwick Tucker
- Brigitte Vallée
- Mark van Hoeij
- Daniel Esteban Vargas Montoya
- Masha Vlasenko
- Benjamin Wesolowski
- James Worrell
- Wadim Zudilin