On the uniform distribution of the sequence $\{\alpha\lambda^x\}$

Let $\lambda>1$ be a real transcendental number. In this paper a number $\alpha$ is constructed such that the sequence $\{\alpha\lambda^x\}_{x=1}^\infty$ is completely uniformly distributed.
For real $\lambda_\nu>1$ ($\nu=1,\dots,s$) numbers $\alpha_1,\dots,\alpha_s$ are constructed such that the remainder of the uniform distribution of the sequence ($\{\alpha_1\lambda_1^x\},\dots,\{\alpha_s\lambda_s^x\}$), $x=\nobreak1,\dots,P$, is equal to $O\bigl(P^{1/2}(\ln P)^{s+1/2}\bigr)$.
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