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Fundamentals of metric theory of real numbers in their \(\overline{Q_3} \)-representation. (English) Zbl 1484.11166

Summary: In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by \(Q_3\)-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.
We have been studied the functions that store the “tails” of \(\overline{Q_3} \)-representation of numbers and the set of such functions, some metric problems and some problems of probability theory are connected with \(\overline{Q_3} \)-representation.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A78 Hausdorff and packing measures
Full Text: DOI

References:

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