Distribution of the Kronecker sequence and related Diophantine problems
by
Nikolai Moshchevitin
Moscow State University

Let \alpha₁, ... , \alpha_s, 1 be linearly independent over Z. The sequence of fractional parts

({\alpha1 k}, ... , {\alphas k} ) in [0;1)s, k = 1, 2, 3, ...

is defined to be Kronecker sequence. Kronecker was the first who proved that this sequence is dense in [0;1)^s.

Let \gamma = (\gamma₁, ... , \gamma_s) be a point in the cube [0;1)^s and put

Np(\gamma) = #{x in N:1 <= x <= p;{\alphaj x } < \gammaj , j = 1, ... , s }

We discuss results dealing with "local" discrepancy D_p (\gamma) = N_p(\gamma)- \gamma₁... \gamma_s p and "global" discrepancy defined by D_p = sup_{\gamma in [0;1)^s}| D_p (\gamma) |. According to the famous H.Weyl's theorem (1916) D_p = o(p), p --> \infty.

By means of so-called "Khinchin singular systems" one can show that for any s >= 2 and for any increasing positive function \psi(y) under condition \psi(y) = o(y), y --> +\infty one can find a vector \alpha with linearly independent together with 1 components \alpha_j such that D_p >> \psi(p) for any natural p.

We shall deal with some kind of ßmooth" F-discrepancy. For Z^s periodic function F(x₁, .., x_s) in C^m ([0, 1]^s), m >= exp(20slogs) we define S(q, j) = \sum_{k = 1}^qF(\alpha₁ k +j₁, ... , \alpha_s k + j_s). In 1998 we proved that for any \alpha₁, ... , \alpha_s linearly independent with 1 and for any j in [0;1)^s one can find a sequence of integers q_\nu --> + \infty such that S(q_\nu , j) = O(1) , \nu --> +\infty. The proof is based on special successive minima of lattices and evaluation of certain quasiperiods of F.

Some related topics and conjectures also will be considered.

Date received: January 24, 2004