# 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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November 12, 2024, 14:00 CET (UTC +1)

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).

November 26, 2024, 14:00 CET (UTC +1)

Haojie Ren (Technion): The dimension of Bernoulli convolutions in R^{d} (paper)

For λ = (λ_{1},...,λ_{d}) ∈ (0,1)^{d} with λ_{1}>...>λ_{d}, denote by μ_{λ} the Bernoulli convolution associated to λ. That is, μ_{λ} is the distribution of the random vector ∑_{n≥0} ±(λ_{1}^{n},...,λ_{d}^{n}), where the ± signs are chosen independently and with equal weight. Assuming for each 1 ≤ j ≤ d that λ_{j} is not a root of a polynomial with coefficients ±1,0, we prove that the dimension of μ_{λ} equals min {dim_{L}μ_{λ},d}, where dim_{L}μ_{λ} is the Lyapunov dimension. This is a joint work with Ariel Rapaport.

December 10, 2024, 13:30 CET (UTC +1)

Ali Messaoudi (São Paulo State University): TBA

(joint talk with Dyadisc 7: Brazilian-chilean and french interplay for symbolic dynamics. An attempt at a low-carbon intercontinental conference)

## Past talks

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 {S_{i} : 1 ≤ i ≤ d} can be concatenated to create a d-dimensional hybrid sequence (S_{1}, . . . , S_{d}). 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 H_{4} (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 H_{4} 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 (q_{n} y) (video) (slides)

The distribution of sequences of the form (q_{n} y) with (q_{n}) 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 (q_{n}) 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 t^{n} 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 R^{n} . 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 R^{2}, 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 R^{2} 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} ε_{i} 2^{n-1-i} where k = ∑_{0≤i<n} ε_{i} 2^{i} with ε_{i} ∈ {0,1}, i ∈ {0,...,n−1}, ε_{n−1} = 1. A natural question is to estimate the number of primes p ∈ [2^{n-1}, 2^{n}) 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 ∈ [2^{n-1}, 2^{n}) 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
(N_{1}, ..., N_{m}). 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.