Rényi β - expansions of 1 with β > 1 a real algebraic number , Perron numbers and a classification problem
نویسنده
چکیده
We prove that for all algebraic number β > 1 the strings of zeros in the Rényi βexpansion dβ(1) of 1 exhibit a lacunarity bounded above by log(s(Pβ))/ log(β), where s(Pβ) is the size of the minimal polynomial of β. The conjecture about the specification of the β-shift, equivalently the uniform discreteness of the sets Zβ of β-integers, for β a Perron number is discussed. We propose a classification of algebraic numbers β > 1 according to the asymptotic “quotient of the gap” value of the Rényi β-expansion of 1 and examplify it, in a complementary classification of Blanchard’s with the classes C1 to C5.
منابع مشابه
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