Functiones et Approximatio Commentarii Mathematici

Two exceptional classes of real numbers

Purusottam Rath

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In a recent paper, Bugeaud and Dubickas have given an explicit characterisation of a rather remarkable class of transcendental numbers which are exceptional from the perspective of distribution of exponential sequences modulo $1$. Muchbefore, Helson and Kahane, from a completely different point-of-view hadexistentially exhibited another class of exceptional real numbers which conjecturally are either rational or transcendental. Wondering whether these two rather large class of real numbers overlap, we study their distribution functions and our investigation gives the first indication that these two interesting class of real numbers originating from different contexts are most likely different. We also frame a natural conjecture in this set up which would establish the above assertion. Our results can be regarded as the first step towards this conjecture.

Article information

Funct. Approx. Comment. Math., Volume 38, Number 1 (2008), 81-91.

First available in Project Euclid: 18 December 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K06: General theory of distribution modulo 1 [See also 11J71]
Secondary: 11K31: Special sequences

exponential sequences distribution functions


Rath, Purusottam. Two exceptional classes of real numbers. Funct. Approx. Comment. Math. 38 (2008), no. 1, 81--91. doi:10.7169/facm/1229624653.

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