Two Distribution Problems for Polynomials

Abstract

Let (x _n ), n = l, 2,… be a sequence of real numbers and denote by A(x _n , N, x) the number of n ≤ N such that the fractional part {x _n } is contained in the given interval [0, x) ⊆ [0,1). The sequence (x _n ) is called uniformly distributed mod 1 if

$$ \mathop{{\lim }}\limits_{{N \to \infty }} \frac{{A\left( {{x_n},N,x} \right)}}{N} = x $$

(1)

for all x ∊ [0, l]. It is well-known that any non-constant polynomial (p(n)), n = l, 2,… is uniformly distributed mod 1 provided that at least one non-constant coefficient is irrational (cf. [9], p. 27). Similar distribution problems are natural also for functions instead of sequences. For a real-valued function f: R₀ ⁺ →R set

$$ A\left( {f(t),\;T,\;x} \right) = \int\limits_0^T {{\chi_{{\left[ {0,\left. x \right)} \right.}}}\left( {\left\{ {f(t)} \right\}} \right)dt;} $$

(2)

(f(t)), t ≥ 0 is called C-uniformly distributed mod 1 if

$$ \mathop{{\lim }}\limits_{{T \to \infty }} \frac{{A\left( {f(t),\;T,\;x} \right)}}{T} = x $$

(3)

for all x ∈ [0,1] (cf. [9], p. 78): χ _E denotes the characteristic function of the set E. In [4] the following result is shown.