arXiv subjects

Christiane Frougny

Publications and source records attributed to Christiane Frougny.

11 recordsLinked to original sources

arXiv2020

The carry propagation of the successor function

Given any numeration system, we call carry propagation at a number $N$ the number of digits that are changed when going from the representation of $N$ to the one of $N+1$, and amortized carry propagation the limit of the mean of the carry propagations at the first $N$ integers, when $N$ 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. We address the problem of the existence of the amortized carry propagation in non-standard numeration systems of various kinds: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. We tackle the problem by three different types of techniques: combinatorial, algebraic, and ergodic. For each kind of numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.

math.CO↗

arXiv2019

On-line algorithms for multiplication and division in real and complex numeration systems

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $β$ such that $|β|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $β$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(β, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $β$ and a digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > |β|$. For a complex base $β$ and symmetric digit set $A$ of contiguous integers, the system $(β, A)$ has the OL Property if $\# A > β\overlineβ + |β+ \overlineβ|$. Provided that addition and subtraction are realizable in parallel in the system $(β, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $β=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $β=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $β= -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + ω$, where $ω= \exp{\frac{2iπ}{3}}$, with digits $A = \{0, \pm 1, \pm ω, \pm ω^2 \}$.

cs.DS↗

arXiv2018

Two applications of the spectrum of numbers

Let the base $β$ be a complex number, $|β|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $β$ is the set $S_{A}(β) = \{\sum_{k=0}^n a_kβ^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$. We show that the spectrum $S_{A}(β)$ has an accumulation point if and only if $0$ has a particular $(β, A)$-representation, said to be \emph{rigid}. The first application is restricted to the case that $β>1 $ and the alphabet is $A=\{-M, \ldots, M\}$, $M \ge 1$ integer. We show that the set $Z_{β,M}$ of infinite $(β, A)$-representations of $0$ is recognizable by a finite Büchi automaton if and only if the spectrum $S_A(β)$ has no accumulation point. Using a result of Akiyama-Komornik and Feng, this implies that $Z_{β, M}$ is recognizable by a finite Büchi automaton for any positive integer $M \ge \lceil β\rceil -1$ if and only if $β$ is a Pisot number. This improves the previous bound $M \ge \lceil β\rceil $. For the second application the base and the digits are complex. We consider the on-line algorithm for division of Trivedi and Ercegovac generalized to a complex numeration system. In on-line arithmetic the operands and results are processed in a digit serial manner, starting with the most significant digit. The divisor must be far from $0$, which means that no prefix of the $(β,A)$-representation of the divisor can be small. The numeration system $(β,A)$ is said to \emph{allow preprocessing} if there exists a finite list of transformations on the divisor which achieve this task. We show that $(β,A )$ allows preprocessing if and only if the spectrum $S_{A}(β)$ has no accumulation point.

math.NT↗

arXiv2016

Minimal digit sets for parallel addition in non-standard numeration systems

We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base $β$ in $\mathbb{C}$ and a finite digit set $\mathcal{A}$ of contiguous integers containing $0$. For a fixed base $β$, we focus on the question of the size of the alphabet allowing to perform addition in constant time independently of the length of representation of the summands. We produce lower bounds on the size of such alphabet $\mathcal{A}$. For several types of well studied bases (negative integer, complex numbers $ -1 + \imath$, $2 \imath$, and $\imath \sqrt{2}$, quadratic Pisot unit, and the non-integer rational base), we give explicit parallel algorithms performing addition in constant time. Moreover we show that digit sets used by these algorithms are the smallest possible.

math.NT↗

arXiv2013

$k$-block parallel addition versus $1$-block parallel addition in non-standard numeration systems

Parallel addition in integer base is used for speeding up multiplication and division algorithms. $k$-block parallel addition has been introduced by Kornerup in 1999: instead of manipulating single digits, one works with blocks of fixed length $k$. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing parallel addition. In this paper, we mainly focus on a certain class of real bases --- the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the $d$-bonacci base, which satisfies the equation $X^d = X^{d-1} + X^{d-2} + \cdots + X + 1$. If in a base being a $d$-bonacci number $1$-block parallel addition is possible on the alphabet $\mathcal{A}$, then $\#\mathcal{A} \geq d+1$; on the other hand, there exists a $k\in\mathbb{N}$ such that $k$-block parallel addition in this base is possible on the alphabet $\{0,1,2\}$, which cannot be reduced. In particular, addition in the Tribonacci base is $14$-block parallel on alphabet $\{0,1,2\}$.

math.NT↗

arXiv2011

Parallel addition in non-standard numeration systems

We consider numeration systems where digits are integers and the base is an algebraic number $β$ such that $|β|>1$ and $β$ satisfies a polynomial where one coefficient is dominant in a certain sense. For this class of bases $β$, we can find an alphabet of signed-digits on which addition is realizable by a parallel algorithm in constant time. This algorithm is a kind of generalization of the one of Avizienis. We also discuss the question of cardinality of the used alphabet, and we are able to modify our algorithm in order to work with a smaller alphabet. We then prove that $β$ satisfies this dominance condition if and only if it has no conjugate of modulus 1. When the base $β$ is the Golden Mean, we further refine the construction to obtain a parallel algorithm on the alphabet $\{-1,0,1\}$. This alphabet cannot be reduced any more.

math.NT↗

arXiv2010

Negative bases and automata

We study expansions in non-integer negative base -β introduced by Ito and Sadahiro. Using countable automata associated with (-β)-expansions, we characterize the case where the (-β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the (-β)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base β to negative base -β. When β is a Pisot number, the conversion can be realized by a finite on-line transducer.

cs.FL↗

arXiv2010

Rational numbers with purely periodic $β$-expansion

We study real numbers $β$ with the curious property that the $β$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $γ(β)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $β$-expansion. We prove that $γ(β)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $η$. This result is motivated by the observation of Akiyama and Scheicher that $γ(η)=0.666 666 666 086 ...$ is surprisingly close to 2/3.

math.NT↗

arXiv2009

Minimal weight expansions in Pisot bases

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $β$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $β$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form.

cs.DM↗

arXiv2008

Univoque numbers and an avatar of Thue-Morse

Univoque numbers are real numbers $λ> 1$ such that the number 1 admits a unique expansion in base $λ$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j λ^{-(j+1)}$, with $a_j \in \{0, 1, ..., \lceil λ\rceil -1\}$ for every $j \geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\{0, 1, >..., b\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \to 31$, $2 \to 30$, $1 \to 03$, $0 \to 02$, occurs in a "universal" manner.

math.NT↗

arXiv2006

Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number β>1 such that the Rényi expansion of 1 is finite, of the form d_β(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_βthat are the fixed point of a substitution associated with a simple Parry number β. It is shown that the word u_βcontains infinitely many palindromes if and only if t_1=t_2= ... =t_{m-1} \geq t_m. Numbers βsatisfying this condition are the so-called confluent Pisot numbers. If t_m=1 then u_βis an Arnoux-Rauzy word. We show that if βis a confluent Pisot number then P(n+1)+ P(n) = C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the number of factors of length n in u_β. We then give a complete description of the set of palindromes, its structure and properties.

math.CO↗