In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-β}$ of numbers whose representation uses only non-negative powers of $-β$, the so-called $(-β)$-integers. We describe the distances between consecutive elements of $\Z_{-β}$. In case that this set is non-trivial we associate to $β$ an infinite word $\boldsymbol{v}_{-β}$ over an (in general infinite) alphabet. The self-similarity of $\Z_{-β}$, i.e., the property $-β\Z_{-β}\subset \Z_{-β}$, allows us to find a morphism under which $\boldsymbol{v}_{-β}$ is invariant. On the example of two cubic irrational bases $β$ we demonstrate the difference between Rauzy fractals generated by $(-β)$-integers and by $β$-integers.
In this article, we propose a variant of the usual Ostrowski $α$-numeration (where $α$ is a real in [0, 1[) that codes integers (positive as well as negative) and reals of [0, 1[ (instead of [--$α$, 1--$α$[), so that for every integer n, n and {n$α$} have the same coding sequence. These coding sequences respect natural lexicographic orders and will be used to prove well known results on order properties of Kronecker sequences ({n$α$ -- $β$}) n .
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 \sq…
Parallel addition, i.e., addition with limited carry propagation, has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions on the alphabet allowing parallel addition. Under certain assumptions, we prove the same lower bound on the size of the generalized alphabet that is known for alphabets consisting of consecutive integers. We also extend the characterization of bases allowing parallel addition to numeration systems with non-integer alphabets.
We consider a numeration system in the ring of integers ${\mathcal O}_K$ of a number field, which we assume to be principal. We prove that the property of being a prime in ${\mathcal O}_K$ is decorrelated from two fundamental examples of automatic sequences relative to the chosen numeration system: the Thue-Morse and the Rudin-Shapiro sequences. This is an analogue, in ${\mathcal O}_K$, of results of Mauduit-Rivat which were concerned with the case $K={\mathbb Q}$.
Institut universitaire de France et IRIF, Université Paris-Cité（法国大学研究院与IRIF，巴黎西岱大学）；Department of Statistical Science, University College London（伦敦大学学院统计科学系）；School of Mathematical Sciences, Queen Mary University of London（伦敦大学玛丽女王学院数学科学学院）
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like numeration of some rationals, simply related to entries a, b, d. So, we obtain quadratic complexity algorithms to compute these invariants. As a consequence, we will prove that, for these numerical semigroups, the type is always lower than the embedding dimension and deduce Wilf's property. We also consider the '' reverse problem'' : gi…
Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven number. Our main result shows that for each degree there exists an infinite set of bases $b$ for which $b$-Niven numbers of that degree exist. The high degree $b$-Niven numbers are given by explicit formulas and have all digits different from zero.
The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbolβ=(β_0,\dots,β_{p-1})$ determining sofic systems. We show that a necessary condition is that the product $δ=\prod_{i=0}^{p-1}β_i$ is an algebraic integer and all of the bases $β_0,\ldots,β_{p-1}$ belong to the algebraic field ${\mathbb Q}(δ)$. On the other hand, we also give a sufficient condition: if $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, then the system associated with the alternate base $\boldsymbolβ=(β_0,\dots,β_{p-1})$ is sofic. The second aim of th…
Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.
Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.
Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.
