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Transactions of the American Mathematical Society

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Digital representations using the greatest integer function


Author: Bruce Reznick
Journal: Trans. Amer. Math. Soc. 312 (1989), 355-375
MSC: Primary 11A63
DOI: https://doi.org/10.1090/S0002-9947-1989-0954602-4
MathSciNet review: 954602
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Abstract: Let $ {S_d}(\alpha)$ denote the set of all integers which can be expressed in the form $ \sum {{\varepsilon _i}[{\alpha ^i}]} $, with $ {\varepsilon _i} \in \{ 0, \ldots ,d - 1\} $, where $ d \geq 2$ is an integer and $ \alpha \geq 1$ is real, and let $ {I_d}$ denote the set of $ \alpha $ so that $ {S_d}(\alpha) = {{\mathbf{Z}}^ + }$. We show that $ {I_d} = [1,{r_d}) \cup \{ d\} $, where $ {r_2} = {13^{1/4}},{r_3} = {22^{1/3}}$ and $ {r_2} = {({d^2} - d - 2)^{1/2}}$ for $ d \geq 4$. If $ \alpha \notin {I_d}$ we show that $ {T_d}(\alpha)$, the complement of $ {S_d}(\alpha)$, is infinite, and discuss the density of $ {T_d}(\alpha)$ when $ \alpha < d$. For $ d \geq 4$ and a particular quadratic irrational $ \beta = \beta (d) < d$, we describe $ {T_d}(\beta)$ explicitly and show that $ \vert{T_d}(\beta) \cap [0,n]\vert$ is of order $ {n^{e(d)}}$, where $ e(d) < 1$.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0954602-4
Article copyright: © Copyright 1989 American Mathematical Society

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