A Diophantine Problem

Abstract

Let $v = \left( {{v_n}} \right), n \geqslant 0$, be an infinite sequence (mod 1). Let p ∈ [1,+∞]. The symbol |..| denotes the ”norm” on the torus R/Z, i.e the distance to the nearest integer. We define the “norm” of the sequence v

$$\left\| v \right\|p = \mathop {\lim \sup }\limits_{N \to \infty } {\left( {\frac{1}{N}\sum\limits_{n - 0}^{N = 1} | {v_n}{|^p}} \right)^{1/p}}for{\text{ }}1 \leqslant p < \infty , $$

and

$$\left\| v \right\| = {\left\| v \right\|_\infty } = \mathop {\lim \sup }\limits_{N \to \infty } |{v_N}| $$

.