Asymptotically uniform distributions

Random vectors $X=(X_1,X_2,\dots,X_s)$ with values in the Euclidean space $\mathbb R^s$ ($s\ge1$) are considered. By definition, the fractional part $\{X\}$ of a vector $X$ is the vector $(\{X_1\},\{X_2\},\dots,\{X_s\})$, with values in the unit cube of $\mathbb R^s$. The deviation of the probability distribution of $\{X\}$ from the uniform distribution is estimated using the Poisson distribution formula. The case of large $s$ is given special attention. Connections with number theory, with questions of random number generation, and with the so-called ‘Benford law’ (the law of the ‘first significant digit’) are discussed.