FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2052550900> ?p ?o ?g. }

• W2052550900 endingPage "860" @default.
• W2052550900 startingPage "841" @default.
• W2052550900 abstract "Given a number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>beta > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the <italic>beta-transformation</italic> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T equals upper T Subscript beta> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>β<!-- β --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>T =T_{beta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of left-bracket 0 comma 1 right-bracket> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>x in [0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T x colon equals beta x> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>x</mml:mi> <mml:mo>:=</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Tx := beta x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (mod 1). The number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be a beta-number if the orbit <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace upper T Superscript n Baseline left-parenthesis 1 right-parenthesis right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{T^{n}(1)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite, hence eventually periodic. In this case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the root of a monic polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>R(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with integer coefficients called the characteristic polynomial of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>P(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the minimal polynomial of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R left-parenthesis x right-parenthesis equals upper P left-parenthesis x right-parenthesis upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>R(x) = P(x)Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is the factor <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which concerns us here in case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be cyclotomic in this case, particularly if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than beta greater-than 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1 > beta > 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 1 comma 1.9324 right-bracket union left-bracket 1.9333 comma 1.96 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1.9324</mml:mn> <mml:mo stretchy=false>]</mml:mo> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>1.9333</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1.96</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[1,1.9324]cup [1.9333,1.96]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (an infinite set), by a search up to degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=50> <mml:semantics> <mml:mn>50</mml:mn> <mml:annotation encoding=application/x-tex>50</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 1.9 comma 2 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>1.9</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[1.9,2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=60> <mml:semantics> <mml:mn>60</mml:mn> <mml:annotation encoding=application/x-tex>60</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 1.96 comma 2 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>1.96</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[1.96,2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and to degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=20> <mml:semantics> <mml:mn>20</mml:mn> <mml:annotation encoding=application/x-tex>20</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 2 comma 2.2 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2.2</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[2,2.2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We find the smallest counterexample, the counterexample of smallest degree, examples where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nonreciprocal, and examples where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from above, and infinite sequences of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nonreciprocal which converge to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from below and to the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=6> <mml:semantics> <mml:mn>6</mml:mn> <mml:annotation encoding=application/x-tex>6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 1 comma 2 right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[1,2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Pisot numbers for which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is cyclotomic are related to an interesting closed set of numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced by Flatto, Lagarias and Poonen in connection with the zeta function of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding=application/x-tex>T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our examples show that the set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Pisot numbers is not a subset of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2052550900 created "2016-06-24" @default.
• W2052550900 creator A5052216098 @default.
• W2052550900 date "1996-01-01" @default.
• W2052550900 modified "2023-10-13" @default.
• W2052550900 title "On beta expansions for Pisot numbers" @default.
• W2052550900 cites W1595543849 @default.
• W2052550900 cites W1597251915 @default.
• W2052550900 cites W1772877864 @default.
• W2052550900 cites W1991450583 @default.
• W2052550900 cites W2050125167 @default.
• W2052550900 cites W2052282156 @default.
• W2052550900 cites W2067657822 @default.
• W2052550900 cites W2068660767 @default.
• W2052550900 cites W2075595154 @default.
• W2052550900 cites W2076789781 @default.
• W2052550900 cites W2078637583 @default.
• W2052550900 cites W2160949125 @default.
• W2052550900 cites W2432093536 @default.
• W2052550900 cites W2901771308 @default.
• W2052550900 cites W4232488423 @default.
• W2052550900 cites W4237941720 @default.
• W2052550900 cites W4246435416 @default.
• W2052550900 cites W4255154153 @default.
• W2052550900 doi "https://doi.org/10.1090/s0025-5718-96-00693-x" @default.
• W2052550900 hasPublicationYear "1996" @default.
• W2052550900 type Work @default.
• W2052550900 sameAs 2052550900 @default.
• W2052550900 citedByCount "22" @default.
• W2052550900 countsByYear W20525509002012 @default.
• W2052550900 countsByYear W20525509002014 @default.
• W2052550900 countsByYear W20525509002015 @default.
• W2052550900 countsByYear W20525509002022 @default.
• W2052550900 crossrefType "journal-article" @default.
• W2052550900 hasAuthorship W2052550900A5052216098 @default.
• W2052550900 hasBestOaLocation W20525509001 @default.
• W2052550900 hasConcept C114614502 @default.
• W2052550900 hasConcept C199360897 @default.
• W2052550900 hasConcept C202444582 @default.
• W2052550900 hasConcept C2776174256 @default.
• W2052550900 hasConcept C33923547 @default.
• W2052550900 hasConcept C41008148 @default.
• W2052550900 hasConceptScore W2052550900C114614502 @default.
• W2052550900 hasConceptScore W2052550900C199360897 @default.
• W2052550900 hasConceptScore W2052550900C202444582 @default.
• W2052550900 hasConceptScore W2052550900C2776174256 @default.
• W2052550900 hasConceptScore W2052550900C33923547 @default.
• W2052550900 hasConceptScore W2052550900C41008148 @default.
• W2052550900 hasIssue "214" @default.
• W2052550900 hasLocation W20525509001 @default.
• W2052550900 hasLocation W20525509002 @default.
• W2052550900 hasOpenAccess W2052550900 @default.
• W2052550900 hasPrimaryLocation W20525509001 @default.
• W2052550900 hasRelatedWork W1587224694 @default.
• W2052550900 hasRelatedWork W1979597421 @default.
• W2052550900 hasRelatedWork W2007980826 @default.
• W2052550900 hasRelatedWork W2061531152 @default.
• W2052550900 hasRelatedWork W2077600819 @default.
• W2052550900 hasRelatedWork W2142036596 @default.
• W2052550900 hasRelatedWork W2911598644 @default.
• W2052550900 hasRelatedWork W3002753104 @default.
• W2052550900 hasRelatedWork W4225152035 @default.
• W2052550900 hasRelatedWork W4245490552 @default.
• W2052550900 hasVolume "65" @default.
• W2052550900 isParatext "false" @default.
• W2052550900 isRetracted "false" @default.
• W2052550900 magId "2052550900" @default.
• W2052550900 workType "article" @default.