Randomness with Respect to the Signed-Digit Representation

The ordinary notion of algorithmic randomness of reals can be characterised as MartinLöf randomness with respect to the Lebesgue measure or as Kolmogorov randomness with respect to the binary representation. In this paper we study the question how the notion of algorithmic randomness induced by the signed-digit representation of the real numbers is related to the ordinary notion of algorithmic randomness. We first consider the image measure on real numbers induced by the signed-digit representation. We call this measure the signed-digit measure and using the Fourier transform of this measure and the Riemann-Lebesgue Lemma we prove that this measure is not absolutely continuous with respect to the Lebesgue measure. We also show that the signed-digit measure can be obtained as a weakly convergent convolution of discrete measures and therefore, by a classical Theorem of Jessen and Wintner the Lebesgue measure is not absolutely continuous with ∗Supported by the South African National Research Foundation (NRF). †Supported by the Austrian Science Foundation (FWF), project S9606, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory.” 2 M. Archibald, V. Brattka, C. Heuberger / Randomness with respect to the Signed-Digit Representation respect to the signed-digit measure. Finally, we provide an invariance theorem which shows that if a computable map preserves Martin-Löf randomness, then its induced image measure has to be absolutely continuous with respect to the target space measure. This theorem can be considered as a loose analog for randomness of the Banach-Mazur theorem for computability. Using this Invariance Theorem we conclude that the notion of randomness induced by the signed-digit representation is incomparable with the ordinary notion of randomness.