# 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.
We use BigBlueButton and Zoom.

If you want to participate in the seminar, please contact Wolfgang Steiner by email to `numeration@irif.fr`

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## Talks in the past

July 14, 2020

Attila Pethő (University of Debrecen): On diophantine properties of generalized number systems - finite and periodic representations (video) (slides)

In this talk we investigate elements with special patterns in their representations in number systems in algebraic number fields. We concentrate on periodicity and on the representation of rational integers. We prove under natural assumptions that there are only finitely many S-units whose representation is periodic with a fixed period. We prove that the same holds for the set of values of polynomials at rational integers.

July 7, 2020

Hajime Kaneko (University of Tsukuba): Analogy of Lagrange spectrum related to geometric progressions (video) (slides) (paper)

Classical Lagrange spectrum is defined by Diophantine approximation properties of arithmetic progressions. The theory of Lagrange spectrum is related to number theory and symbolic dynamics. In our talk we introduce significantly analogous results of Lagrange spectrum in uniform distribution theory of geometric progressions. In particular, we discuss the geometric sequences whose common ratios are Pisot numbers. For studying the fractional parts of geometric sequences, we introduce certain numeration system.

This talk is based on a joint work with Shigeki Akiyama.

June 30, 2020

Niels Langeveld (Leiden University): Continued fractions with two non integer digits (video) (slides)

In this talk, we will look at a family of continued fraction expansions for which the digits in the expansions can attain two different (typically non-integer) values, named α_{1} and α_{2} with α_{1}α_{2} ≤ 1/2 . If α_{1}α_{2} < 1/2 we can associate a dynamical system to these expansions with a switch region and therefore with lazy and greedy expansions. We will explore the parameter space and highlight certain values for which we can construct the natural extension (such as a family for which the lowest digit cannot be followed by itself). We end the talk with a list of open problems.

June 23, 2020

Derong Kong (Chongqing University): Univoque bases of real numbers: local dimension, Devil's staircase and isolated points (video) (slides) (paper)

Given a positive integer M and a real number x, let U(x) be the set of all bases q in (1,M+1] such that x has a unique q-expansion with respect to the alphabet {0,1,...,M}. We will investigate the local dimension of U(x) and prove a 'variation principle' for unique non-integer base expansions. We will also determine the critical values and the topological structure of U(x).

June 16, 2020

Carlos Matheus (CNRS, École Polytechnique): Approximations of the Lagrange and Markov spectra (video) (slides) (journal) (arXiv)

The Lagrange and Markov spectra are closed subsets of the positive real numbers defined in terms of diophantine approximations. Their topological structures are quite involved: they begin with an explicit discrete subset accumulating at 3, they end with a half-infinite ray of the form [4.52...,∞), and the portions between 3 and 4.52... contain complicated Cantor sets. In this talk, we describe polynomial time algorithms to approximate (in Hausdorff topology) these spectra.

June 9, 2020

Simon Baker (University of Birmingham): Equidistribution results for self-similar measures (video) (slides) (paper)

A well known theorem due to Koksma states that for Lebesgue almost every x>1 the sequence (x^{n}) is uniformly distributed modulo one. In this talk I will discuss an analogue of this statement that holds for fractal measures. As a corollary of this result we show that if C is equal to the middle third Cantor set and t≥1, then almost every x in C+t is such that (x^{n}) is uniformly distributed modulo one. Here almost every is with respect to the natural measure on C+t.

June 2, 2020

Henna Koivusalo (University of Vienna): Linear repetition in polytopal cut and project sets (video) (slides)

Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study repetition of finite patterns, where sets with linear pattern repetition can be considered as the most ordered aperiodic sets.

Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In an earlier work it was shown that for cut and project sets with a cube window, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the set has minimal complexity and (ii) the irrational slope satisfies a certain Diophantine condition. In a new joint work with Jamie Walton, we give a generalisation of this result for other polytopal windows, under mild geometric conditions. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.

May 26, 2020

Célia Cisternino (University of Liège): Ergodic behavior of transformations associated with alternate base expansions (video) (slides)

We consider a p-tuple of real numbers greater than 1, 𝛃=(𝛽_{1},…,𝛽_{p}), called an alternate base, to represent real numbers. Since these representations generalize the 𝛽-representation introduced by Rényi in 1958, a lot of questions arise. In this talk, we will study the transformation generating the alternate base expansions (greedy representations). First, we will compare the 𝛃-expansion and the (𝛽_{1}*…*𝛽_{p})-expansion over a particular digit set and study the cases when the equality holds. Next, we will talk about the existence of a measure equivalent to Lebesgue, invariant for the transformation corresponding to the alternate base and also about the ergodicity of this transformation.

This is a joint work with Émilie Charlier and Karma Dajani.

May 19, 2020

Boris Solomyak (University of Bar-Ilan): On singular substitution Z-actions (video) (slides) (paper)

We consider primitive aperiodic substitutions on d letters and the spectral properties of associated dynamical systems. In an earlier work we introduced a *spectral cocycle*, related to a kind of matrix Riesz product, which extends the (transpose) substitution matrix to the d-dimensional torus. The asymptotic properties of this cocycle provide local information on the (fractal) dimension of spectral measures. In the talk I will discuss a sufficient condition for the singularity of the spectrum in terms of the top Lyapunov exponent of this cocycle.

This is a joint work with A. Bufetov.

May 12, 2020

Olivier Carton (Université de Paris): Preservation of normality by selection (video) (slides)

We first recall Agafonov's theorem which states that finite state selection preserves normality. We also give two slight extensions of this result to non-oblivious selection and suffix selection. We also propose a similar statement in the more general setting of shifts of finite type by defining selections which are compatible with the shift.

May 5, 2020

Narad Rampersad (University of Winnipeg): Ostrowski numeration and repetitions in words (video) (slides)

One of the classical results in combinatorics on words is Dejean's Theorem, which specifies the smallest exponent of repetitions that are avoidable on a given alphabet. One can ask if it is possible to determine this quantity (called the *repetition threshold*) for certain families of infinite words. For example, it is known that the repetition threshold for Sturmian words is 2+phi, and this value is reached by the Fibonacci word. Recently, this problem has been studied for *balanced words* (which generalize Sturmian words) and *rich words*. The infinite words constructed to resolve this problem can be defined in terms of the Ostrowski-numeration system for certain continued-fraction expansions. They can be viewed as *Ostrowski-automatic* sequences, where we generalize the notion of *k-automatic sequence* from the base-k numeration system to the Ostrowski numeration system.