On the distribution of sequences connected with digit-representation (Q1104359)

Let \(1=b(0)<b(1)<b(2)<..\). be a sequence of integers, then for every positive integer N there is an s with \(b(s)\leq N<b(s+1)\) and a unique e(s)\(\in {\mathbb{N}}\) with \(N=e(s)b(s)+N_{s-1}\) and with \(0\leq N_{s- 1}<b(s)\). This leads to a unique representation of the positive integers N with respect to the sequence b. For a wide class of digit depending functions \(F: {\mathbb{N}}_ 0\to {\mathbb{Z}}\) necessary and sufficient conditions are established such that the sequence (F(N)\(\cdot x)\), \(N=1,2,..\). is uniformly distributed modulo 1; x denotes a given real number.