Missouri Journal of Mathematical Sciences

Fraction Sets for Basic Digit Sets

Darren D. Wick

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A finite set of integers $D$ with $0\in D$ is basic for the base $b\in \mathbb Z$ if every $z\in\mathbb Z$ can be written uniquely as an integer in the base $b$ with digits from $D$. Since such a base representation is unsigned, basic sets must have a negative base $b$ or some negative integers in $D$. The fraction set for $(b,D)$ is the set of all representable numbers with integer part zero. We show that if $(b,D)$ is basic and $D$ consists of consecutive digits, then the fraction set is an interval of length one whose endpoints have redundant representations. Furthermore, we show that if $D$ does not consist of consecutive integers, then $F$ is disconnected.

Article information

Missouri J. Math. Sci., Volume 22, Issue 2 (2010), 91-96.

First available in Project Euclid: 1 August 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16}


Wick, Darren D. Fraction Sets for Basic Digit Sets. Missouri J. Math. Sci. 22 (2010), no. 2, 91--96. doi:10.35834/mjms/1312233138. https://projecteuclid.org/euclid.mjms/1312233138

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