Fibonacci, van der Corput and Riesz-Nágy. (English) Zbl 1227.11032
Summary: What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system – also known as Zeckendorf’s system – we pose ourselves the problem of extending it in a natural way to represent all real numbers in \((0,1)\). We see that this natural extension leads to what is known as the \(\phi\)-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the van der Corput sequence in \((0,1)\), a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity. Lastly, combining the Fibonacci system and the binary in a very special way we obtain a singular function, more specifically, the inverse of one of the family of Riesz-Nágy.
MSC:
| 11A63 | Radix representation; digital problems |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
| 26A30 | Singular functions, Cantor functions, functions with other special properties |
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