## Powers of rationals modulo 1 and rational base number systems

Citations: | 15 - 2 self |

### BibTeX

@MISC{Akiyama_powersof,

author = {Shigeki Akiyama and Christiane Frougny and Jacques Sakarovitch},

title = {Powers of rationals modulo 1 and rational base number systems},

year = {}

}

### OpenURL

### Abstract

### Citations

2194 | The Art of computer programming - Knuth - 1981 |

632 | The Encyclopedia of Integer Sequences
- Sloane, Plouffe
- 1995
(Show Context)
Citation Context ...Example 3 For p q 3 = 2 , the constant ω 3 2 11, 24]. Its decimal expansion: ω 3 2 is the constant K(3) already discussed in [17, =1.622270502884767315956950982 ··· is recorded as Sequence A083286 in =-=[23]-=-. Observe that, in the same case, the sequence (Gk)k�1 is Sequence A061419 in [23]. ⋄ 20s5 Representation of the reals Every infinite word a in A N is given a real value x by π: and a is called a p q ... |

619 |
Combinatorics on Words
- Lothaire
- 1983
(Show Context)
Citation Context ...bers of [0, 1[ is shift-invariant, and its closure forms a symbolic dynamical system called the β-shift. The properties of the β-shift are well understood, using the so-called “β-expansion of 1”, see =-=[18, 14]-=-. When β is a Pisot number 2 ,theβ number system shares many properties with the integer base case: the set of greedy representations is recognizable by a finite automaton; the conversion between two ... |

280 |
Introduction to Automata Theory
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...ry representation of an n divisible by 3 and g is the binary representation of n/3, possibly prefixed with some 0’s. For more definitions and results on automata theory the reader is referred to [7], =-=[12]-=-, or [22], to quote a few. Three more things should be added though. First, the 5stransitions of the transducers we shall consider are labeled in A×B —wecallthese transducers letter-to-letter. If one ... |

183 |
On the β-expansions of real numbers
- Parry
- 1960
(Show Context)
Citation Context ...bers of [0, 1[ is shift-invariant, and its closure forms a symbolic dynamical system called the β-shift. The properties of the β-shift are well understood, using the so-called “β-expansion of 1”, see =-=[18, 14]-=-. When β is a Pisot number 2 ,theβ number system shares many properties with the integer base case: the set of greedy representations is recognizable by a finite automaton; the conversion between two ... |

179 |
Signed-digit number representation for fast parallel arithmetic
- Avizienis
- 1961
(Show Context)
Citation Context ... the literature: [13, Vol. 2, Chap. 4] or [14, Chap. 7], for instance, give extensive references. Representation in integer base with signed digits was popularized in computer arithmetic by Avizienis =-=[2]-=- and can even be found earlier in a work of Cauchy [4]. When the base is a real number β > 1, any non-negative real number is given an expansion on the canonical alphabet {0, 1,...,⌊β⌋} by the greedy ... |

176 |
Representations for real numbers and their ergodic properties
- Rényi
- 1957
(Show Context)
Citation Context ...und earlier in a work of Cauchy [4]. When the base is a real number β > 1, any non-negative real number is given an expansion on the canonical alphabet {0, 1,...,⌊β⌋} by the greedy algorithm of Rényi =-=[21]-=-; a number may have several β-representations on the canonical alphabet, but the greedy one is the greatest in the lexicographical order. The set of greedy β-expansions of numbers of [0, 1[ is shift-i... |

56 | Systems of numeration - Fraenkel - 1985 |

55 |
Infinite Words
- Perrin, Pin
- 2004
(Show Context)
Citation Context ... W p q is the set of infinite words w in AN such that any prefix of w is in 0∗ L p q .As0∗Lp q is prefix-closed —sinceL p q is prefix-closed and the empty word belongs to L p q — W p q is closed (see =-=[19]-=-) in the compact set AN , hence compact. Since π is continuous, X p q Suppose that [0, ω p q ] \ X p q is closed. is a non-empty open set, containing a real u. Let y = sup{x ∈ X p q | x<u} and z =inf{... |

43 |
Representation of numbers and finite automata
- Frougny
- 1992
(Show Context)
Citation Context ...e integer base case: the set of greedy representations is recognizable by a finite automaton; the conversion between two alphabet of digits (in particular addition) is realized by a finite transducer =-=[10]-=-. -expansion of an integer N :itisawayofwritingN by an algorithm which produces least significant digits first. We prove: In this work, we first define the p q in the base p q Theorem 1 Every non-nega... |

30 |
Eléments de théorie des automates, Vuibert (2003). English translation: Elements of Automata Theory
- Sakarovitch
(Show Context)
Citation Context ...entation of an n divisible by 3 and g is the binary representation of n/3, possibly prefixed with some 0’s. For more definitions and results on automata theory the reader is referred to [7], [12], or =-=[22]-=-, to quote a few. Three more things should be added though. First, the 5stransitions of the transducers we shall consider are labeled in A×B —wecallthese transducers letter-to-letter. If one retains t... |

20 | Eléments de théorie des automates, Vuibert - Sakarovitch - 2003 |

15 |
An unsolved problem on the powers of 3/2
- Mahler
(Show Context)
Citation Context ...in [16] and he goes on: “Pisot, Vijayaraghavan and Andr Weil did however show that there are infinitely many limit points.” (cf. [25] for instance.) With this problem as a background, Mahler asked in =-=[15]-=- whether there exists a non zero real z such that the fractional part of z (3/2) n for n =0, 1,... fall into [0, 1/2[. It is not known whether such a real — called Z-number — does exist but Mahler sho... |

14 | Functional Iteration and the Josephus Problem
- Odlyzko, Wilf
- 1991
(Show Context)
Citation Context ...he relations of the p q -expansions of reals with other problems in combinatorics and number theory. The first one is the so-called “Josephus problem” in which a certain constant K(p) is defined (cf. =-=[17, 11, 24]-=-) which is a special case of our constant ω p (with q = p − 1) and this definition yields a new q method for computing K(p). The connection with the second problem, namely the distribution of the powe... |

7 |
On the range of fractional parts {ξ(p/q
- Flatto, Lagarias, et al.
- 1995
(Show Context)
Citation Context ... such that Z p (I) is empty and conversely the search for subsets I as small q as possible such that Z p (I) is non-empty. q Along the first line, remarkable progress��has been made by Flatto et al. (=-=[8]-=-) who proved that the set of reals s such that Z p s, s + q 1 �� p is empty is dense in [0, 1 − 1 p ], and recently Bugeaud [3] proved that its complement �� is �� of Lebesgue measure 0. Along 4 61 th... |

6 |
On the distance from a rational power to the nearest integer
- Dubickas
(Show Context)
Citation Context ... and the closest integer. Corollary 4 assures that there are (infinitely many) positive numbers x such that ||x(3/2) n || < 1/3 forn =0, 1,.... This is to be compared with a recent result of Dubickas =-=[6]-=- who showed that ||x(3/2) n || < 0.238117 ...(n =0, 1,...) implies that x =0 — hence extending his result [5] on the distribution of {xαn } which works basically for any algebraic number α. Though the... |

6 | The Josephus Problem
- Halbeisen, Hungerbühler
- 1997
(Show Context)
Citation Context ...he relations of the p q -expansions of reals with other problems in combinatorics and number theory. The first one is the so-called “Josephus problem” in which a certain constant K(p) is defined (cf. =-=[17, 11, 24]-=-) which is a special case of our constant ω p (with q = p − 1) and this definition yields a new q method for computing K(p). The connection with the second problem, namely the distribution of the powe... |

5 |
mod one transformations and the distribution of fractional parts {ξ(p/q
- Bugeaud, Linear
(Show Context)
Citation Context ...ong the first line, remarkable progress��has been made by Flatto et al. ([8]) who proved that the set of reals s such that Z p s, s + q 1 �� p is empty is dense in [0, 1 − 1 p ], and recently Bugeaud =-=[3]-=- proved that its complement �� is �� of Lebesgue measure 0. Along 4 61 the other line, Pollington [20] showed that Z 3 2 65 , 65 is non-empty. Our contribution to the problem can be seen as an improve... |

5 |
Arithmetical properties of powers of algebraic numbers
- Dubickas
(Show Context)
Citation Context ...|x(3/2) n || < 1/3 forn =0, 1,.... This is to be compared with a recent result of Dubickas [6] who showed that ||x(3/2) n || < 0.238117 ...(n =0, 1,...) implies that x =0 — hence extending his result =-=[5]-=- on the distribution of {xαn } which works basically for any algebraic number α. Though there is a distance between 1/3 and0.238117 ...,we expect that our Y p q is minimal in the sense that for any pr... |

4 |
Sur les moyens d’éviter les erreurs dans les calculs numériques
- Cauchy
(Show Context)
Citation Context ...], for instance, give extensive references. Representation in integer base with signed digits was popularized in computer arithmetic by Avizienis [2] and can even be found earlier in a work of Cauchy =-=[4]-=-. When the base is a real number β > 1, any non-negative real number is given an expansion on the canonical alphabet {0, 1,...,⌊β⌋} by the greedy algorithm of Rényi [21]; a number may have several β-r... |

4 |
Remarks and problems on finite and periodic continued fractions, Enseign
- France
- 1993
(Show Context)
Citation Context ...er, indeed the problem of proving whether they form a dense set or not, is a frustrating question: “Thisveryold problem of Pisot and Vijayaraghavan is still unanswered.” writes Michel Mends France in =-=[16]-=- and he goes on: “Pisot, Vijayaraghavan and Andr Weil did however show that there are infinitely many limit points.” (cf. [25] for instance.) With this problem as a background, Mahler asked in [15] wh... |

3 |
On a Sequence Related to the Josephus Problem
- Stephan
- 2003
(Show Context)
Citation Context ...he relations of the p q -expansions of reals with other problems in combinatorics and number theory. The first one is the so-called “Josephus problem” in which a certain constant K(p) is defined (cf. =-=[17, 11, 24]-=-) which is a special case of our constant ω p (with q = p − 1) and this definition yields a new q method for computing K(p). The connection with the second problem, namely the distribution of the powe... |

3 |
On the fractional parts of the powers of a number
- Vijayaraghavan
- 1940
(Show Context)
Citation Context ...ot and Vijayaraghavan is still unanswered.” writes Michel Mends France in [16] and he goes on: “Pisot, Vijayaraghavan and Andr Weil did however show that there are infinitely many limit points.” (cf. =-=[25]-=- for instance.) With this problem as a background, Mahler asked in [15] whether there exists a non zero real z such that the fractional part of z (3/2) n for n =0, 1,... fall into [0, 1/2[. It is not ... |

2 |
Progressions arithmétiques généralisées et le problème des (3/2) n , Comptes Rendus de l’Académie des Sciences Série A 292
- Pollington
- 1981
(Show Context)
Citation Context ... of reals s such that Z p s, s + q 1 �� p is empty is dense in [0, 1 − 1 p ], and recently Bugeaud [3] proved that its complement �� is �� of Lebesgue measure 0. Along 4 61 the other line, Pollington =-=[20]-=- showed that Z 3 2 65 , 65 is non-empty. Our contribution to the problem can be seen as an improvement of this last result. Theorem 3 If p � 2q − 1 ,thereexistsasubsetYp � � q is countably infinite. s... |

1 |
Mahler’s Z-number and 3/2-number system, preprint
- Akiyama
(Show Context)
Citation Context ... to describe this connection, let us first set the framework of this deeply intriguing problem. 3 Koksma proved that for almost every real number θ > 1 the sequence {θn } is uniformely distributed in =-=[0, 1]-=- , but very few results are known for specific values of θ. One of these is that if θ is a Pisot number, then the above sequence converges to 0 if we identify [0, 1[ with R/Z. �� �n� p Experimental re... |

1 |
Power fractional parts
- Weisstein
(Show Context)
Citation Context ...esults show that the distribution of q for coprime positive integers p>q� 2 looks more “chaotic” than the distribution of the fractional part of the powers of a transcendental number like e or π (cf. =-=[26]-=-). The next step in attacking this problem has been to fix the rational p q and to study the distribution of the sequence � � � p n� fn(z) = z q is the same, is finite. This is a very remarkable featu... |

1 | On the range of fractional parts {ξ(p/q) n }, Acta Arithmetica 70 - Flatto, Lagarias, et al. - 1995 |

1 | Systems of numeration, The American Mathematical Monthly 92 - Fraenkel - 1985 |

1 | Representation of numbers and finite automata, Mathematical Systems Theory 25 - Frougny - 1992 |

1 | The Josephus problem, Journal de Théorie des Nombres de Bordeaux 9 - Halbeisen, Hungerbüler - 1997 |