Digital representations using the greatest integer function

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Let Sd(α) denote the set of all integers which can be expressed in the form Σ ε[αi] with εi € (0,…, d ™ 1), where d ≥ 2 is an integer and α ≥ 1 is real, and let ld denote the set of a so that Sd(α) = Z+. We show that Id = [1, rd]∪(d), where r2 = 13l/4, r3 = 221/3 and r4 = (d2 − d − 2)l/2 for d ≥ 4. If α Ε Id, we show that Td(α), the complement of Sd(α), is infinite, and discuss the density of Td(α) when α < d. For d ≥ 4 and a particular quadratic irrational β = β(d) < d, we describe Td(β) explicitly and show that ΙTd(β) ∩ [0, n]Ι is of order ne(d), where e(d) > 1.

Original languageEnglish (US)
Pages (from-to)355-375
Number of pages21
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - Mar 1989

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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