One World Numeration Seminar
This is an online seminar on numeration systems and related topics (see the series of Numeration conferences), in the spirit of other One World Seminars; talks are on Zoom.
If you want to participate in the seminar, please contact the organisers (Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner) by email to numeration@irif.fr
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February 18, 2025, 14:00 CET (UTC +1)
Neil MacVicar (Queen's University): Intersecting Cantor sets generated by Complex Radix Expansions (slides) (paper)
Consider the classical middle third Cantor set. This is a self-similar set containing all the numbers in the unit interval which have a ternary expansion that avoids the digit 1. We can ask when the intersection of the Cantor set with a translate of itself is also self-similar. Sufficient and necessary conditions were given by Deng, He, and Wen in 2008. This question has also been generalized to classes of subsets of the unit interval. I plan to discuss how existing ideas can be used to address the question for certain self-similar sets with dimension greater than one. These ideas will be illustrated using a class of self-similar sets in the plane that can be realized as radix expansions in base -n+i where n is a positive integer. I will also discuss a property of the fractal dimensions of these kinds of intersections.
March 18, 2025, 14:00 CET (UTC +1)
Valentin Ovsienko (CNRS, Université de Reims-Champagne-Ardenne): From Catalan numbers to integrable dynamics: continued fractions and Hankel determinants for q-numbers (paper1) (paper2) (paper3)
The classical Catalan and Motzkin numbers have remarkable continued fraction expansions, the corresponding sequences of Hankel determinants consist of -1, 0 and 1 only. We find an infinite family of power series corresponding to q-deformed real numbers that have very similar properties. Moreover, their sequences of Hankel determinants turn out to satisfy Somos and Gale-Robinson recurrences. (Partially based on a joint work with Emmanuel Pedon.)
April 1, 2025, 14:00 CEST (UTC +2)
Meng Wu (Oulun yliopisto): TBA
April 15, 2025, 14:00 CEST (UTC +2)
James Cumberbatch (Purdue University): TBA
April 29, 2025, 14:00 CEST (UTC +2)
Yan Huang (Chongqing University): TBA
May 13, 2025, 14:00 CEST (UTC +2)
Artem Dudko (IM PAN): TBA
May 27, 2025, 14:00 CEST (UTC +2)
Savinien Kreczman (Université de Liège): TBA
June 10, 2025, 14:00 CEST (UTC +2)
Yuta Suzuki (Rikkyo University): TBA
Past talks
February 4, 2025
Giulia Salvatori (Politecnico di Torino): Continued Fractions, Quadratic Forms, and Regulator Computation for Integer Factorization (video) (slides) (paper)
In the realm of integer factorization, certain methods, such as CFRAC, leverage the properties of continued fractions, while others, like SQUFOF, combine these properties with the tools provided by quadratic forms. Recently, Michele Elia revisited the fundamental concepts of SQUFOF, including reduced quadratic forms, distance between quadratic forms, and Gauss composition, offering a new perspective for designing factorization methods.
In this seminar, we present our algorithm, which is a refinement of Elia’s method, along with a precise analysis of its computational cost. Our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of Q(√N). The computation of the regulator governs the total computational cost, which is subexponential, and in particular O(exp(3/√8 (ln N ln ln N)1/2)). This makes our method more efficient than CFRAC and SQUFOF, though less efficient than the General Number Field Sieve.
We identify a broad family of integers to which our method is applicable including certain classes of RSA moduli. Finally, we introduce some promising avenues for refining our method. These span several areas, ranging from Algebraic Number Theory, particularly for estimating the size of the regulator of Q(√N), to Analytic Number Theory, particularly for computing a specific class of L-functions.
Joint work with Nadir Murru
January 21, 2025
Thomas Garrity (Williams College): Multi-dimensional continued fractions and integer partitions: Using the Natural Extension to create a tree structure on partitions (video) (slides) (paper1) (paper2) (paper3)
The goal is to explore the recently found interplay between integer partitions and the dynamics of the triangle map (a type of multi-dimensional continued fraction algorithm). (This is joint work with Baalbaki, Bonanno, Del Vigna and Isola.)
As we will see, the triangle map gives an almost internal symmetry from the set of integer partitions to itself, which in turn allows the generation of any number of new partition identities.
Further, this allows us to place a tree structure on the space of all integer partitions. (This is joint work with Joe Fox and with Jacob Lehmann Duke). This tree structure allowed us to find the natural extension of the triangle map in any dimension. As with the classical Farey map, the dynamics of this map, in every dimension, has an indifferent fixed point, which in turn can be used to understand the structure of the integer partition tree.
Among the many different types of multi-dimensional continued fractions that exist, for still unknown reasons it appears that the triangle map is the only one that is "partition" compatible.
Thus we use the triangle map (stemming from number theory and dynamics) to understand classical integer partition numbers from combinatorics, and use partition numbers to understand the dynamics of the triangle map.
January 7, 2025
Jean-Paul Allouche (CNRS, IMJ-PRG, Sorbonne Université): Kolam, Ethnomathematics, and Morphisms (video) (slides) (paper1) (paper2)
Kolam is a form of traditional decorative art in India, that is drawn by using rice flour, white stone powder, chalk or chalk powder. It is often practised by women in front of their house entrance. One particular kolam uses a 4x4 grid. It can also be found in the Vanuatu Islands, in Africa, etc. We show that a natural generalization on grids of size 8x8, 16x16, etc. is linked to... the Thue-Morse sequence. Further we unveil two (twin) morphisms that generate this family of kolam, and show that they appear in unrelated and somewhat unexpected fields. Time permitting we will allude to a vast, relatively new, field: ethnomathematic(s).
December 10, 2024
Ali Messaoudi (Universidade Estadual Paulista): Adding machine, automata and Julia sets (slides)
(joint talk with Dyadisc 7: Brazilian-chilean and french interplay for symbolic dynamics. An attempt at a low-carbon intercontinental conference)
A stochastic adding machine (defined by P.R. Killeen and T.J. Taylor) is a Markov chain whose states are natural integers, which models the process of adding the number 1 but where there is a probability of failure in which a carry is not performed when necessary. In this lecture, we will talk about dynamical, spectral and probabilistic properties of extensions for the stochastic adding machine and their connections with other topics as Julia sets, Automata and Dynamical Systems on Banach spaces.
November 26, 2024
Haojie Ren (Technion): The dimension of Bernoulli convolutions in Rd (video) (slides) (paper)
For λ = (λ1,...,λd) ∈ (0,1)d with λ1>...>λd, denote by μλ the Bernoulli convolution associated to λ. That is, μλ is the distribution of the random vector ∑n≥0 ±(λ1n,...,λdn), where the ± signs are chosen independently and with equal weight. Assuming for each 1 ≤ j ≤ d that λj is not a root of a polynomial with coefficients ±1,0, we prove that the dimension of μλ equals min {dimLμλ,d}, where dimLμλ is the Lyapunov dimension. This is a joint work with Ariel Rapaport.
November 12, 2024
Florian Luca (Stellenbosch University): On a question of Douglass and Ono (video) (slides) (paper)
It is known that the partition function p(n) obeys Benford's law in any integer base b ≥ 2. A similar result was obtained by Douglass and Ono for the plane partition function PL(n) in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base b ≥ 2 and string f of digits in base b they asked for an explicit value N(b,f) such that there exists n ≤ N(b,f) with the property that PL(n) starts with the string f when written in base b. In my talk, I will present an explicit value for N(b,f) both for the partition function p(n) as well as for the plane partition function PL(n).
October 29, 2024
Victor Shirandami (University of Manchester): Probabilistic Effectivity in the Subspace Theorem and the Distribution of Algebraic Projective Points (video) (slides)
The celebrated Roth’s theorem in Diophantine Approximation determines the degree to which an algebraic number may be approximated by rationals. A corollary of this theorem yields a transcendence criterion for real numbers based off of their decimal expansion. This theorem, and its broad generalisation due to Schmidt, famously suffers from ineffectivity. This motivates one to address this issue in the probabilistic context, whereby one makes progress in the direction of effectivity in an appropriately defined probabilistic regime. From this analysis is derived an analogue of Khintchine's theorem for algebraic numbers, answering a question of Beresnevich, Bernick, and Dodson on a density version of Waldschmidt’s conjecture.
October 15, 2024
Steven Robertson (University of Manchester): Low Discrepancy Digital Hybrid Sequences and the t-adic Littlewood Conjecture (video) (slides) (paper)
The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number d, any collection of one-dimensional so-called low discrepancy sequences {Si : 1 ≤ i ≤ d} can be concatenated to create a d-dimensional hybrid sequence (S1, . . . , Sd). Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. In this talk, an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences is provided.
Specifically, it is shown that any counterexample to the so-called t-adic Littlewood Conjecture (t-LC) can be used to create a low discrepancy digital Kronecker-Van der Corput sequence. Such counterexamples to t-LC are known explicitly over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the Robertson. All necessary concepts will be defined in the talk.
October 1, 2024, 14:00 CEST (UTC +2)
Dong Han Kim (Dongguk University): Uniform Diophantine approximation on the Hecke group H4 (video) (slides)
Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation. We study uniform Diophantine approximation properties on the Hecke group H4 in terms of the Rosen continued fractions. For a given real number α, the best approximations are convergents of the Rosen continued fraction and the dual Rosen continued fraction of α. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants. This is joint work with Ayreena Bakhtawar and Seul Bee Lee.
September 17, 2024
Simon Kristensen (Aarhus Universitet): On the distribution of sequences of the form (qn y) (video) (slides)
The distribution of sequences of the form (qn y) with (qn) a sequence of integers and y a real number have attracted quite a bit of attention, for instance due to their relation to inhomogeneous Littlewood type problems. In this talk, we will provide some results on the Lebesgue measure and Hausdorff dimension on the set of points in the unit interval approximated to a certain rate by points from such a sequence. A feature of our approach is that we obtain estimates even in the case when the sequence (qn) grows rather slowly. This is joint work with Tomas Persson.
June 18, 2024
Noy Soffer Aranov (Technion) : Escape of Mass of the Thue Morse Sequence (video) (slides)
One way to study the distribution of quadratic number fields is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational θ, the degrees of the periodic part of the continued fraction of tn θ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We show that the Thue Morse sequence is a counterexample to their conjecture. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing work joint with Erez Nesharim.
May 21, 2024
Gaétan Guillot (Université Paris-Saclay): Approximation of linear subspaces by rational linear subspaces (video)
We elaborate on a problem raised by Schmidt in 1967: rational approximation of linear subspaces of Rn . In order to study the quality approximation of irrational numbers by rational ones, one can introduce the exponent of irrationality of a number. We can then generalize this notion in the framework of vector subspaces for the approximation of a subspace by so-called rational subspaces.
After briefly introducing the tools for constructing this generalization, I will present the different possible studies of this object. Finally I will explain how we can construct spaces with prescribed exponents.
May 7, 2024
Tom Kempton (University of Manchester): The Dynamics of the Fibonacci Partition Function (video) (slides) (paper)
The Fibonacci partition function R(n) counts the number of ways of representing a natural number n as the sum of distinct Fibonacci numbers. For example, R(6)=2 since 6=5+1 and 6=3+2+1. An explicit formula for R(n) was recently given by Chow and Slattery. In this talk we express R(n) in terms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of R(n).
April 23, 2024
Shunsuke Usuki (Kyoto University): On a lower bound of the number of integers in Littlewood's conjecture (video) (slides) (paper1) (paper2)
Littlewood's conjecture is a famous and long-standing open problem which states that, for every (α,β) in R2, n ||nα|| ||nβ|| can be arbitrarily small for some integer n. This problem is closely related to the action of diagonal matrices on SL(3,R)/SL(3,Z), and a groundbreaking result was shown by Einsiedler, Katok and Lindenstrauss from the measure rigidity for this action, saying that Littlewood's conjecture is true except on a set of Hausdorff dimension zero. In this talk, I will explain about a new quantitative result on Littlewood's conjecture which gives, for every (α,β) in R2 except on sets of small Hausdorff dimension, an estimate of the number of integers n which make n ||nα|| ||nβ|| small. The keys for the proof are the measure rigidity and further studies on behavior of empirical measures for the diagonal action.
March 26, 2024
Nikita Shulga (La Trobe University): Radical bound for Zaremba’s conjecture (video) (slides) (paper)
Zaremba's conjecture states that for each positive integer q, there exists a coprime integer a, smaller than q, such that partial quotients in the continued fraction expansion of a/q are bounded by some absolute constant. Despite major breakthroughs in the recent years, the conjecture is still open. In this talk I will discuss a new result towards Zaremba's conjecture, proving that for each denominator, one can find a numerator, such that partial quotients are bounded by the radical of the denominator, i.e. the product of distinct prime factors. This generalizes the result by Niederreiter and improves upon some results of Moshchevitin-Murphy-Shkredov.
March 12, 2024
Joël Ouaknine (Max Planck Institute for Software Systems): The Skolem Landscape (video) (slides)
The Skolem Problem asks how to determine algorithmically whether a given linear recurrence sequence (such as the Fibonacci numbers) has a zero. It is a central question in dynamical systems and number theory, and has many connections to other branches of mathematics and computer science. Unfortunately, its decidability has been open for nearly a century! In this talk, I will present a survey of what is known on the Skolem Problem and related questions, including recent and ongoing developments.
February 13, 2024
Bartosz Sobolewski (Jagiellonian University in Kraków and Montanuniversität Leoben): Block occurrences in the binary expansion of n and n+t (video) (slides) (paper)
Let s(n) denote the sum of binary digits of a nonnegative integer n. In 2012 Cusick asked whether for every nonnegative integer t the set of n satisfying s(n+t) ≥ s(n) has natural density strictly greater than 1/2. So far it is known that the answer is affirmative for almost all t (in the sense of density) and s(n+t)-s(n) has approximately Gaussian distribution. During the talk we consider an analogue of this problem concerning the function r(n), which counts the occurrences of the block 11 in the binary expansion of n. In particular, we prove that the distribution of r(n+t)-r(n) is approximately Gaussian as well. We also discuss a generalization to an arbitrary block of binary digits. This is a joint work with Lukas Spiegelhofer.
January 30, 2024
Cathy Swaenepoel (Université Paris Cité): Reversible primes (video) (slides) (paper)
The properties of the digits of prime numbers and various other sequences of integers have attracted great interest in recent years.
For any positive integer k, we denote by rev(k) the reverse of k in base 2, defined by rev(k) = ∑0≤i<n εi 2n-1-i where k = ∑0≤i<n εi 2i with εi ∈ {0,1}, i ∈ {0,...,n−1}, εn−1 = 1. A natural question is to estimate the number of primes p ∈ [2n-1, 2n) such that rev(p) is prime. We will present a result which provides an upper bound of the expected order of magnitude. Our method is based on a sieve argument and also allows us to obtain a strong lower bound for the number of integers k such that k and rev(k) have at most 8 prime factors (counted with multiplicity). We will also present an asymptotic formula for the number of integers k ∈ [2n-1, 2n) such that k and rev(k) are squarefree.
This is a joint work with Cécile Dartyge, Bruno Martin, Joël Rivat and Igor Shparlinski.
January 16, 2024
Karma Dajani (Universiteit Utrecht): Alternating N-continued fraction expansions (video) (slides)
We introduce a family of maps generating continued fractions where the digit 1 in the numerator is replaced cyclically by some given non-negative integers (N1, ..., Nm). We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration. This is joint work with Niels Langeveld.