# Numeration — Archive

## Archive of One World Numeration Seminar talks in 2020

December 15, 2020
Lukas Spiegelhofer (Montanuniversität Leoben): The digits of n+t   (video) (slides) (paper)

We study the binary sum-of-digits function s2 under addition of a constant t. For each integer k, we are interested in the asymptotic density δ(k,t) of integers t such that s2(n+t) - s2(n) = k. In this talk, we consider the following two questions.
(1) Do we have ct = δ(0,t) + δ(1,t) + ... > 1/2? This is a conjecture due to T. W. Cusick (2011).
(2) What does the probability distribution defined by k → δ(k,t) look like?
We prove that indeed ct > 1/2 if the binary expansion of t contains at least M blocks of contiguous ones, where M is effective. Our second theorem states that δ(j,t) usually behaves like a normal distribution, which extends a result by Emme and Hubert (2018).
This is joint work with Michael Wallner (TU Wien).

December 8, 2020
Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa): Limit theorems on counting large continued fraction digits   (video) (slides) (journal) (arXiv)

We establish a central limit theorem for counting large continued fraction digits (an), that is, we count occurrences {an>bn}, where (bn) is a sequence of positive integers. Our result improves a similar result by Philipp, which additionally assumes that bn tends to infinity. Moreover, we also show this kind of central limit theorem for counting the number of occurrences entries such that the continued fraction entry lies between dn and dn(1+1/cn) for given sequences (cn) and (dn). For such intervals we also give a refinement of the famous Borel–Bernstein theorem regarding the event that the nth continued fraction digit lying infinitely often in this interval. As a side result, we explicitly determine the first φ-mixing coefficient for the Gauss system - a result we actually need to improve Philipp's theorem. This is joint work with Marc Kesseböhmer.

December 1, 2020
Michael Barnsley (Australian National University): Rigid fractal tilings   (video) (slides) (paper)

I will describe recent work, joint with Louisa Barnsley and Andrew Vince, concerning a symbolic approach to self-similar tilings. This approach uses graph-directed iterated function systems to analyze both classical tilings and also generalized tilings of what may be unbounded fractal subsets of Rn. A notion of rigid tiling systems is defined. Our key theorem states that when the system is rigid, all the conjugacies of the tilings can be described explicitly. In the seminar I hope to prove this for the case of standard IFSs.

November 17, 2020
Jacques Sakarovitch (Irif, CNRS  and  Télécom Paris): The carry propagation of the successor function   (video) (slides) (paper)

Given any numeration system, the carry propagation at an integer N is the number of digits that change between the representation of N and N+1. The carry propagation of the numeration system as a whole is the average carry propagations at the first N integers, as N tends to infinity, if this limit exists.
In the case of the usual base p numeration system, it can be shown that the limit indeed exists and is equal to p/(p-1). We recover a similar value for those numeration systems we consider and for which the limit exists.
The problem is less the computation of the carry propagation than the proof of its existence. We address it for various kinds of numeration systems: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. This problem is tackled with three different types of techniques: combinatorial, algebraic, and ergodic, each of them being relevant for different kinds of numeration systems.
This work has been published in Advances in Applied Mathematics 120 (2020). In this talk, we shall focus on the algebraic and ergodic methods.
Joint work with V. Berthé (Irif), Ch. Frougny (Irif), and M. Rigo (Univ. Liège).

November 10, 2020
Pieter Allaart (University of North Texas): On the smallest base in which a number has a unique expansion   (video) (slides) (paper)

For x>0, let U(x) denote the set of bases q in (1,2] such that x has a unique expansion in base q over the alphabet {0,1}, and let f(x)=inf U(x). I will explain that the function f(x) has a very complicated structure: it is highly discontinuous and has infinitely many infinite level sets. I will describe an algorithm for numerically computing f(x) that often gives the exact value in just a small finite number of steps. The Komornik-Loreti constant, which is f(1), will play a central role in this talk. This is joint work with Derong Kong, and builds on previous work by Kong (Acta Math. Hungar. 150(1):194-208, 2016).

November 3, 2020
Tomáš Vávra (University of Waterloo): Distinct unit generated number fields and finiteness in number systems   (video) (slides)

A distinct unit generated field is a number field K such that every algebraic integer of the field is a sum of distinct units. In 2015, Dombek, Masáková, and Ziegler studied totally complex quartic fields, leaving 8 cases unresolved. Because in this case there is only one fundamental unit u, their method involved the study of finiteness in positional number systems with base u and digits arising from the roots of unity in K.
First, we consider a more general problem of positional representations with base beta with an arbitrary digit alphabet D. We will show that it is decidable whether a given pair (𝛽, D) allows eventually periodic or finite representations of elements of OK.
We are then able to prove the conjecture that the 8 remaining cases indeed are distinct unit generated.

October 27, 2020
Mélodie Andrieu (Aix-Marseille University): A Rauzy fractal unbounded in all directions of the plane   (slides)

Until 2001 it was believed that, as for Sturmian words, the imbalance of Arnoux-Rauzy words was bounded - or at least finite. Cassaigne, Ferenczi and Zamboni disproved this conjecture by constructing an Arnoux-Rauzy word with infinite imbalance, i.e. a word whose broken line deviates regularly and further and further from its average direction. Today, we hardly know anything about the geometrical and topological properties of these unbalanced Rauzy fractals. The Oseledets theorem suggests that these fractals are contained in a strip of the plane: indeed, if the Lyapunov exponents of the matricial product associated with the word exist, one of these exponents at least is nonpositive since their sum equals zero. This talk aims at disproving this belief.

October 20, 2020
Paul Surer (University of Natural Resources and Life Sciences, Vienna): Representations for complex numbers with integer digits   (video) (slides)

In this talk we present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

October 13, 2020
Kan Jiang (Ningbo University): Representations of real numbers on fractal sets   (video) (slides)

There are many approaches which can represent real numbers. For instance, the β-expansions, the continued fraction and so forth. Representations of real numbers on fractal sets were pioneered by H. Steinhaus who proved in 1917 that C+C=[0,2] and C−C=[−1,1], where C is the middle-third Cantor set. Equivalently, for any x ∈ [0,2], there exist some y,z ∈ C such that x=y+z. In this talk, I will introduce similar results in terms of some fractal sets.

October 6, 2020
Francesco Veneziano (University of Genova): Finiteness and periodicity of continued fractions over quadratic number fields   (video) (slides) (paper)

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field K; a particular example of these continued fractions is the β-continued fraction introduced by Bernat. We show that for any quadratic Perron number β, the β-continued fraction expansion of elements in Q(β) is either finite of eventually periodic. We also show that for certain four quadratic Perron numbers β, the β-continued fraction represents finitely all elements of the quadratic field Q(β), thus answering questions of Rosen and Bernat.
Based on a joint work with Zuzana Masáková and Tomáš Vávra.

September 29, 2020
Marta Maggioni (Leiden University): Random matching for random interval maps   (video) (slides) (paper)

In this talk we extend the notion of matching for deterministic transformations to random matching for random interval maps. For a large class of piecewise affine random systems of the interval, we prove that this property of random matching implies that any invariant density of a stationary measure is piecewise constant. We provide examples of random matching for a variety of families of random dynamical systems, that includes generalised beta-transformations, continued fraction maps and a family of random maps producing signed binary expansions. We finally apply the property of random matching and its consequences to this family to study minimal weight expansions.
Based on a joint work with Karma Dajani and Charlene Kalle.

September 22, 2020
Yotam Smilansky (Rutgers University): Multiscale Substitution Tilings   (video) (slides) (paper)

Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical system, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results.
Based on joint work with Yaar Solomon.

September 15, 2020
Jeffrey Shallit (University of Waterloo): Lazy Ostrowski Numeration and Sturmian Words   (video) (slides) (paper)

In this talk I will discuss a new connection between the so-called "lazy Ostrowski" numeration system, and periods of the prefixes of Sturmian characteristic words. I will also give a relationship between periods and the so-called "initial critical exponent". This builds on work of Frid, Berthé-Holton-Zamboni, Epifanio-Frougny-Gabriele-Mignosi, and others, and is joint work with Narad Rampersad and Daniel Gabric.

September 8, 2020
Bing Li (South China University of Technology): Some fractal problems in beta-expansions   (video) (slides)

For greedy beta-expansions, we study some fractal sets of real numbers whose orbits under beta-transformation share some common properties. For example, the partial sum of the greedy beta-expansion converges with the same order, the orbit is not dense, the orbit is always far from that of another point etc. The usual tool is to approximate the beta-transformation dynamical system by Markov subsystems. We also discuss the similar problems for intermediate beta-expansions.

September 1, 2020
Bill Mance (Adam Mickiewicz University in Poznań): Hotspot Lemmas for Noncompact Spaces   (video) (slides)

We will explore a correction of several previously claimed generalizations of the classical hotspot lemma. Specifically, there is a common mistake that has been repeated in proofs going back more than 50 years. Corrected versions of these theorems are increasingly important as there has been more work in recent years focused on studying various generalizations of the concept of a normal number to numeration systems with infinite digit sets (for example, various continued fraction expansions, the Lüroth series expansion and its generalizations, and so on). Also, highlighting this (elementary) mistake may be helpful for those looking to study these numeration systems further and wishing to avoid some common pitfalls.

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 α1 and α2 with α1α2 ≤ 1/2 . If α1α2 < 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 (xn) 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 (xn) 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=(𝛽1,…,𝛽p), 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 (𝛽1*…*𝛽p)-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.