Rational base number systems for<i>p</i>-adic numbers

Frougny, Christiane; Klouda, Karel

doi:10.1051/ita/2011114

Cited by 13 publications

(27 citation statements)

References 9 publications

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“…17:3 ON P/Q-RECOGNISABLE SETS 12:7 . In [FK12], the author define them differently. Let us stress that the two definitions are not equivalent, although they define two objects that are close.…”

Section: 33mentioning

confidence: 99%

“…Each has properties that the other has not, hence one object is more convenient than the other depending on the context. To make things precise, we give below the evaluation of a word u = a k−1 • • • a 0 in (A p ) * , as defined in [FK12].…”

Section: 33mentioning

confidence: 99%

“…Rep [FK12] (0) = ε ∀m ∈ N \ {0} Rep [FK12] (m) = Rep [FK12] (n) • a where n ∈ N, a ∈ A p q m = pn + qa 12:8…”

Section: 33mentioning

confidence: 99%

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On p/q-recognisable sets

Marsault¹

2021

Logical Methods in Computer Science

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Let p/q be a rational number. Numeration in base p/q is defined by a function that evaluates each finite word over A_p={0,1,...,p-1} to some rational number. We let N_p/q denote the image of this evaluation function. In particular, N_p/q contains all nonnegative integers and the literature on base p/q usually focuses on the set of words that are evaluated to nonnegative integers; it is a rather chaotic language which is not context-free. On the contrary, we study here the subsets of (N_p/q)^d that are p/q-recognisable, i.e. realised by finite automata over (A_p)^d. First, we give a characterisation of these sets as those definable in a first-order logic, similar to the one given by the B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, we show that the natural order relation and the modulo-q operator are not p/q-recognisable.

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Section: 33mentioning

confidence: 99%

Section: 33mentioning

confidence: 99%

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On p/q-recognisable sets

Marsault¹

2021

Logical Methods in Computer Science

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“…−1, as a p-adic expansion using the same residue class representative set R, we have −1 = ∞ i=0 (p − 1) p i , as an element in the ring Zp of p-adic integers. It is known that for any a ∈ Z, the set of rational integers, the p-adic expansion of a is either finite or periodic [6,16].…”

Section: Introductionmentioning

confidence: 99%

On þ-adic Expansions of Algebraic Integers

Chen

Huang

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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It is well known that every rational integer has a finite or periodic p-adic expansion. In this paper a more general notion of p-adic expansion is introduced for algebraic integers, where given a number field K and a principal prime ideal p in K, a different choice of generator for p is allowed in each stage of the expansion. With the notion of p-adic expansion, we prove that there is always a finite or periodic p-adic expansion for every algebraic integer. Moreover, we prove a bound on the periodicity of the p-adic expansion that depends only on the number field K and the prime ideal p. The proof yields an algorithm for constructing such a p-adic expansion for elements in the ring O of algebraic integers of K, through finding an approximation to the closest vector on the lattice spanned by the unit group of O.As a special case we prove that, similar to rational integers, Gaussian integers are finite or periodic not only in p-adic expansion but also in π-adic expansion, where a fixed generator π for p is used in each stage of the expansion. Moreover, the time complexity of finding a π-adic expansion for a Gaussian integer is polynomial in the length of input, the period, and p, where p is the rational prime contained in p. We implement the algorithm for some quadratic number fields and provide examples which illustrate that the p-adic expansion of the elements in O is either finite or periodic.

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“…One way of thinking about how integer bases such as these work, invented by Propp [9] and popularized by Tanton [10], is the idea of exploding dots, which allows a natural extension into fractional bases. A more rigorous discussion of such fractional bases is covered in [2] and also in [7]. We explain exploding dots and base 3/2 in detail below.…”

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confidence: 99%