On the Number of Optimal Base 2 Representations of Integers

Authors:
Peter J. Grabner;Clemens Heuberger
Affiliations:
Institut für Mathematik A, Technische Universität Graz, Graz, Austria 8010;Institut für Mathematik B, Technische Universität Graz, Graz, Austria 8010
Venue:
Designs, Codes and Cryptography
Year:
2006

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Abstract

We study representations of integers n in binary expansions using the digits 0,卤1. We analyze the average number of such representations of minimal "weight" (= number of non-zero digits). The asymptotic main term of this average involves a periodically oscillating function, which is analyzed in some detail. The main tool is the construction of a measure on [驴1,1], which encodes the number of representations.