Primitive Substitutive Numbers are Closed under Rational Multiplication

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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed ... continued below

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iv, 17 leaves

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Ketkar, Pallavi S. (Pallavi Subhash) August 1998.

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  • Ketkar, Pallavi S. (Pallavi Subhash)

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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.

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iv, 17 leaves

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  • August 1998

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  • March 24, 2014, 8:07 p.m.

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  • Sept. 18, 2014, 1:55 p.m.

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Ketkar, Pallavi S. (Pallavi Subhash). Primitive Substitutive Numbers are Closed under Rational Multiplication, thesis, August 1998; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc278637/: accessed August 19, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .