Unique Representations of Real Numbers in Non-Integer Bases
نویسندگان
چکیده
منابع مشابه
On the Expansions of Real Numbers in Two Integer Bases
Let r ≥ 2 and s ≥ 2 be distinct integers. We establish that, if r and s are multiplicatively independent, then the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0, 1, . . . , r− 1} and {0, 1, . . . , s− 1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words. We also discuss the c...
متن کاملOn the Number of Unique Expansions in Non-integer Bases
Let q > 1 be a real number and let m = m(q) be the largest integer smaller than q. It is well known that each number x ∈ Jq := [0, P ∞ i=1 mq ] can be written as x = P ∞ i=1 ciq −i with integer coefficients 0 ≤ ci < q. If q is a non-integer, then almost every x ∈ Jq has continuum many expansions of this form. In this note we consider some properties of the set Uq consisting of numbers x ∈ Jq ha...
متن کاملUnique Expansions of Real Numbers
It was discovered some years ago that there exist non-integer real numbers q > 1 for which only one sequence (ci) of integers 0 ≤ ci < q satisfies the equality P∞ i=1 ciq −i = 1. The set U of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. For each fix...
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The signed-bit representation of real numbers is like the binary representation, but in addition to 0 and 1 you can also use −1. It lends itself especially well to the constructive (intuitionistic) theory of the real numbers. The first part of the paper develops and studies the signed-bit equivalents of three common notions of a real number: Dedekind cuts, Cauchy sequences, and regular sequence...
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ژورنال
عنوان ژورنال: Mathematical Research Letters
سال: 2001
ISSN: 1073-2780,1945-001X
DOI: 10.4310/mrl.2001.v8.n4.a12