Analysis of Alternative Digit Sets for Nonadjacent Representations

• Autores: Clemens Heuberger, Helmut Prodinger
• Localización: Monatshefte für mathematik, ISSN 0026-9255, Vol. 147, Nº 3, 2006, págs. 219-248
• Idioma: inglés
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• Resumen

  • It is known that every positive integer n can be represented as a finite sum of the form ?iai2i, where ai ? {0, 1,-1} and no two consecutive ai¿s are non-zero (¿nonadjacent form¿, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research.

    The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or -1