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- W2344503516 abstract "Abstract. In [7], it was shown that Euclidean rings R of imaginary qua-dratic integers admit unitary number systems. In this paper, as an appli-cation of the result, we obtain all self similar tiles arising from the unitarynumber systems of R. 1. IntroductionLet Rbe a ring of imaginary quadratic integers, which consists of all algebraicintegers in the eld Qpdfor some square-free negative integer d, and let bbe an element of Rwhose norm N(b) is greater than one. Then a completerepresentatives Dof R=(b) is called a digit set for a base b, where (b) denotesan ideal generated by b. Then a unitary number system (b;D) in Ris de nedso that all non-zero elements of digits in Dconsist of units of R.From [7] we see that among the nine principal ideal domains of imaginaryquadratic integers, only the ve ringsZp 1; Zp 2; Z1 +p 32; Z1 +p 72; Z1 + 112;equipped with Euclidean functions de ned by norms admit unitary numbersystems. As proposed in [7], the concept of a unitary number system can be re-placement of a ‘universal side divisor’ by Motzkin. Furthermore unitary numbersystems are already available in various applications of complex based numbersystems. In particular, Khmelnik [3] and Penny [6] independently showed thata number system with a base b= 1 +p 1 and a digit set D= f0;1ginZp 1yields so called the twin dragon (see Figure 2 in Appendix). Knuth[5] proposed another binary number system with a base b=p 2 and a digit" @default.
- W2344503516 created "2016-06-24" @default.
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- W2344503516 date "2015-09-30" @default.
- W2344503516 modified "2023-09-26" @default.
- W2344503516 title "SELF SIMILAR TILES ARISING FROM THE UNITARY NUMBER SYSTEMS IN EUCLIDEAN RINGS OF IMAGINARY QUADRATIC INTEGERS" @default.
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- W2344503516 doi "https://doi.org/10.7858/eamj.2015.048" @default.
- W2344503516 hasPublicationYear "2015" @default.
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