On α-greedy expansions of numbers

We study a redundant binary number system that was recently introduced by Székely and Wang. For a natural number n, it is defined as follows: let k satisfy ; then k2 is subtracted from n, and the expansion continues recursively. It stops, when a power of 2 is reached.For this and more general number systems, where the factor 2/3 is replaced by a general one, we find an explicit formula for the kth digit εk∈{0,1,2}. This allows us to compute the cumulative frequency of a given digit, among the first N integers. Delange's method produces not only the leading term of order NlogN, but also the fluctuating term of order N, and the Fourier coefficients of the periodic functions that are involved.Furthermore, we can compute the expansions from right to left, by translating the ordinary binary expansion using a (finite state) transducer, provided the factor (such as 2/3) is rational. In this case, we prove that the periodic function mentioned above is nowhere differentiable.