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• W3105844981 endingPage "6249" @default.
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• W3105844981 abstract "Given a real number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>x>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we determine <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript s Baseline left-parenthesis x right-parenthesis colon-equal inf script upper U left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≔</mml:mo> <mml:mi>inf</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>U</mml:mi> </mml:mrow> </mml:mrow> <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_s(x)≔operatorname {inf}{mathscr {U}}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper U left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>U</mml:mi> </mml:mrow> </mml:mrow> <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>{mathscr {U}}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of all bases <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q element-of left-parenthesis 1 comma 2 right-bracket> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <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>qin (1,2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a unique expansion of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding=application/x-tex>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s. We give an explicit description of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript s Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <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_s(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for several regions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-values. For others, we present an efficient algorithm to determine <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript s Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <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_s(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the lexicographically smallest unique expansion of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the infimum is attained for almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript s> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript s> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A large part of the paper is devoted to the level sets <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L left-parenthesis q right-parenthesis colon-equal StartSet x greater-than 0 colon q Subscript s Baseline left-parenthesis x right-parenthesis equals q EndSet> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≔</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L(q)≔{x>0:q_s(x)=q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L left-parenthesis q right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite for almost every <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper K upper L Baseline equals min script upper U left-parenthesis 1 right-parenthesis almost-equals 1.787> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>min</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>U</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≈<!-- ≈ --></mml:mo> <mml:mn>1.787</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>q_{KL}=operatorname {min}{mathscr {U}}(1)approx 1.787</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L left-parenthesis q Subscript upper K upper L Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L(q_{KL})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has both infinitely many left- and infinitely many right accumulation points." @default.
• W3105844981 created "2020-11-23" @default.
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• W3105844981 date "2021-06-07" @default.
• W3105844981 modified "2023-09-30" @default.
• W3105844981 title "On the smallest base in which a number has a unique expansion" @default.
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