On distribution of fractional parts for a system of linear functions. (English. Russian original) Zbl 0719.11037

Moshchevitin, N. G.

Mosc. Univ. Math. Bull. 45, No. 4, 26-30 (1990); translation from Vestn. Mosk. Univ., Ser. I 1990, No. 4, 26-31 (1990).

Let \(\alpha_ 1x,...,\alpha_ sx\) be \(s\geq 1\) real functions, where 1, \(\alpha_ 1,...,\alpha_ s\) form a linearly independent system over the ring of rational integers. Let \(\gamma_ 1,...,\gamma_ s\in (0,1)\) and \(N_ q=N_ q(\gamma_ 1,...,\gamma_ s)\) be the number of integers x satisfying the inequalities \(\{\alpha_ ix\}<\gamma_ i\) (1\(\leq i\leq s)\), \(1\leq x\leq q\), where \(\{\alpha_ ix\}\) denotes the distance of the real number \(\alpha_ ix\) from the nearest integer which do not exceed \(\alpha_ ix.\)
The hypothesis states that there exist constants C and \(\epsilon\) such that the inequality \[ \sup_{\gamma_ 1,...,\gamma_ s}| N_ q- \gamma_ 1...\gamma_ sq| \leq Cq^{1-\epsilon} \] has an infinity of integer solutions q. The author proves this hypothesis under certain conditions on \(\alpha_ 1,...,\alpha_ s\).

Reviewer: S.Kotov (Minsk)

MSC:

11J54	Small fractional parts of polynomials and generalizations
11J25	Diophantine inequalities
11K60	Diophantine approximation in probabilistic number theory
11J70	Continued fractions and generalizations