Powers of rationals modulo 1 and rational base number systems View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2008-09-19

AUTHORS

Shigeki Akiyama, Christiane Frougny, Jacques Sakarovitch

ABSTRACT

A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{p} {q} $$\end{document} allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers. More... »

PAGES

53

References to SciGraph publications

  • 1992-03. Representations of numbers and finite automata in THEORY OF COMPUTING SYSTEMS
  • 1960-09. On theβ-expansions of real numbers in ACTA MATHEMATICA HUNGARICA
  • 1957-09. Representations for real numbers and their ergodic properties in ACTA MATHEMATICA HUNGARICA
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s11856-008-1056-4

    DOI

    http://dx.doi.org/10.1007/s11856-008-1056-4

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1020982046


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Department of Mathematics, Niigata University, Niigata, Japan", 
              "id": "http://www.grid.ac/institutes/grid.260975.f", 
              "name": [
                "Department of Mathematics, Niigata University, Niigata, Japan"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Akiyama", 
            "givenName": "Shigeki", 
            "id": "sg:person.011153327405.03", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011153327405.03"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Universit\u00e9 Paris 8, Paris, France", 
              "id": "http://www.grid.ac/institutes/grid.15878.33", 
              "name": [
                "LIAFA, UMR 7089 CNRS, Paris, France", 
                "Universit\u00e9 Paris 8, Paris, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Frougny", 
            "givenName": "Christiane", 
            "id": "sg:person.011327100311.34", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011327100311.34"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "LTCI, UMR 5141, CNRS / ENST, 46, rue Barrault, 75634, Paris Cedex 13, France", 
              "id": "http://www.grid.ac/institutes/grid.464001.7", 
              "name": [
                "LTCI, UMR 5141, CNRS / ENST, 46, rue Barrault, 75634, Paris Cedex 13, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Sakarovitch", 
            "givenName": "Jacques", 
            "id": "sg:person.010412054043.09", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010412054043.09"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf02020954", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1018173239", 
              "https://doi.org/10.1007/bf02020954"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01368783", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1026715244", 
              "https://doi.org/10.1007/bf01368783"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02020331", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1048516797", 
              "https://doi.org/10.1007/bf02020331"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2008-09-19", 
        "datePublishedReg": "2008-09-19", 
        "description": "A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called \u201cJosephus problem.\u201d More important, these expansions in the base \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\n\\tfrac{p}\n{q}\n$$\\end{document} allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s11856-008-1056-4", 
        "inLanguage": "en", 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1136632", 
            "issn": [
              "0021-2172", 
              "1565-8511"
            ], 
            "name": "Israel Journal of Mathematics", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "168"
          }
        ], 
        "keywords": [
          "real numbers", 
          "countable infinite number", 
          "number theory", 
          "infinite number", 
          "rational numbers", 
          "Base Number System", 
          "modulo 1", 
          "positive integer", 
          "rational base number systems", 
          "integers", 
          "fractional part", 
          "set of expansions", 
          "number system", 
          "Josephus problem", 
          "problem", 
          "such expansion", 
          "combinatorics", 
          "new method", 
          "expansion", 
          "theory", 
          "new interpretation", 
          "regular languages", 
          "number", 
          "unique expansion", 
          "Real", 
          "power", 
          "first example", 
          "set", 
          "distribution", 
          "system", 
          "rational base", 
          "letter", 
          "interpretation", 
          "results", 
          "transducer", 
          "base", 
          "progress", 
          "digits", 
          "part", 
          "addition", 
          "language", 
          "rights", 
          "example", 
          "method"
        ], 
        "name": "Powers of rationals modulo 1 and rational base number systems", 
        "pagination": "53", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1020982046"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s11856-008-1056-4"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s11856-008-1056-4", 
          "https://app.dimensions.ai/details/publication/pub.1020982046"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-05-10T09:59", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220509/entities/gbq_results/article/article_463.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s11856-008-1056-4"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s11856-008-1056-4'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s11856-008-1056-4'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s11856-008-1056-4'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s11856-008-1056-4'


     

    This table displays all metadata directly associated to this object as RDF triples.

    135 TRIPLES      22 PREDICATES      72 URIs      61 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s11856-008-1056-4 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author Nb0398fe4461b40ef89ccc5aed30ab2f8
    4 schema:citation sg:pub.10.1007/bf01368783
    5 sg:pub.10.1007/bf02020331
    6 sg:pub.10.1007/bf02020954
    7 schema:datePublished 2008-09-19
    8 schema:datePublishedReg 2008-09-19
    9 schema:description A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{p} {q} $$\end{document} allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.
    10 schema:genre article
    11 schema:inLanguage en
    12 schema:isAccessibleForFree false
    13 schema:isPartOf N253c05a3aa4341858a67b56b70400676
    14 Nd973746f03d84433aa36ec3711f3b0fd
    15 sg:journal.1136632
    16 schema:keywords Base Number System
    17 Josephus problem
    18 Real
    19 addition
    20 base
    21 combinatorics
    22 countable infinite number
    23 digits
    24 distribution
    25 example
    26 expansion
    27 first example
    28 fractional part
    29 infinite number
    30 integers
    31 interpretation
    32 language
    33 letter
    34 method
    35 modulo 1
    36 new interpretation
    37 new method
    38 number
    39 number system
    40 number theory
    41 part
    42 positive integer
    43 power
    44 problem
    45 progress
    46 rational base
    47 rational base number systems
    48 rational numbers
    49 real numbers
    50 regular languages
    51 results
    52 rights
    53 set
    54 set of expansions
    55 such expansion
    56 system
    57 theory
    58 transducer
    59 unique expansion
    60 schema:name Powers of rationals modulo 1 and rational base number systems
    61 schema:pagination 53
    62 schema:productId Nbf038d3180bb42829e996603fbd749e4
    63 Ne4622648a2d24f8da0bdd28e87de7507
    64 schema:sameAs https://app.dimensions.ai/details/publication/pub.1020982046
    65 https://doi.org/10.1007/s11856-008-1056-4
    66 schema:sdDatePublished 2022-05-10T09:59
    67 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    68 schema:sdPublisher N889c32dae3474ef7a0f7d2720a176e1a
    69 schema:url https://doi.org/10.1007/s11856-008-1056-4
    70 sgo:license sg:explorer/license/
    71 sgo:sdDataset articles
    72 rdf:type schema:ScholarlyArticle
    73 N253c05a3aa4341858a67b56b70400676 schema:volumeNumber 168
    74 rdf:type schema:PublicationVolume
    75 N2ce93774a23e47eeb42cdfaf578dbf5f rdf:first sg:person.011327100311.34
    76 rdf:rest Nc10c648927dc47559bd95bccb854e7f9
    77 N889c32dae3474ef7a0f7d2720a176e1a schema:name Springer Nature - SN SciGraph project
    78 rdf:type schema:Organization
    79 Nb0398fe4461b40ef89ccc5aed30ab2f8 rdf:first sg:person.011153327405.03
    80 rdf:rest N2ce93774a23e47eeb42cdfaf578dbf5f
    81 Nbf038d3180bb42829e996603fbd749e4 schema:name dimensions_id
    82 schema:value pub.1020982046
    83 rdf:type schema:PropertyValue
    84 Nc10c648927dc47559bd95bccb854e7f9 rdf:first sg:person.010412054043.09
    85 rdf:rest rdf:nil
    86 Nd973746f03d84433aa36ec3711f3b0fd schema:issueNumber 1
    87 rdf:type schema:PublicationIssue
    88 Ne4622648a2d24f8da0bdd28e87de7507 schema:name doi
    89 schema:value 10.1007/s11856-008-1056-4
    90 rdf:type schema:PropertyValue
    91 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    92 schema:name Mathematical Sciences
    93 rdf:type schema:DefinedTerm
    94 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    95 schema:name Pure Mathematics
    96 rdf:type schema:DefinedTerm
    97 sg:journal.1136632 schema:issn 0021-2172
    98 1565-8511
    99 schema:name Israel Journal of Mathematics
    100 schema:publisher Springer Nature
    101 rdf:type schema:Periodical
    102 sg:person.010412054043.09 schema:affiliation grid-institutes:grid.464001.7
    103 schema:familyName Sakarovitch
    104 schema:givenName Jacques
    105 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010412054043.09
    106 rdf:type schema:Person
    107 sg:person.011153327405.03 schema:affiliation grid-institutes:grid.260975.f
    108 schema:familyName Akiyama
    109 schema:givenName Shigeki
    110 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011153327405.03
    111 rdf:type schema:Person
    112 sg:person.011327100311.34 schema:affiliation grid-institutes:grid.15878.33
    113 schema:familyName Frougny
    114 schema:givenName Christiane
    115 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011327100311.34
    116 rdf:type schema:Person
    117 sg:pub.10.1007/bf01368783 schema:sameAs https://app.dimensions.ai/details/publication/pub.1026715244
    118 https://doi.org/10.1007/bf01368783
    119 rdf:type schema:CreativeWork
    120 sg:pub.10.1007/bf02020331 schema:sameAs https://app.dimensions.ai/details/publication/pub.1048516797
    121 https://doi.org/10.1007/bf02020331
    122 rdf:type schema:CreativeWork
    123 sg:pub.10.1007/bf02020954 schema:sameAs https://app.dimensions.ai/details/publication/pub.1018173239
    124 https://doi.org/10.1007/bf02020954
    125 rdf:type schema:CreativeWork
    126 grid-institutes:grid.15878.33 schema:alternateName Université Paris 8, Paris, France
    127 schema:name LIAFA, UMR 7089 CNRS, Paris, France
    128 Université Paris 8, Paris, France
    129 rdf:type schema:Organization
    130 grid-institutes:grid.260975.f schema:alternateName Department of Mathematics, Niigata University, Niigata, Japan
    131 schema:name Department of Mathematics, Niigata University, Niigata, Japan
    132 rdf:type schema:Organization
    133 grid-institutes:grid.464001.7 schema:alternateName LTCI, UMR 5141, CNRS / ENST, 46, rue Barrault, 75634, Paris Cedex 13, France
    134 schema:name LTCI, UMR 5141, CNRS / ENST, 46, rue Barrault, 75634, Paris Cedex 13, France
    135 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...