Distribution of the fractional parts of random vectors: The Gaussian case. I

The paper considers a fractional distribution of an $s$-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula and some facts from the theory of representation of integers by square forms. The main attention of this part of the paper is devoted to the case of small values of $s$. The case of large values of $s$ will be consider additionally.