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The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong to$L$^{∞}($T$) may be one-to-one identified with Toeplitz matrices from$B$($l$_{2}) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:$ms$—a unital commutative subalgebra of$l$^{∞},$C$($l$_{2}) the matricial analogue of the space of all continuous periodic functions$C$($T$), the matricial Haar polynomials, etc.

We are studying the Diophantine exponent μ_{$n$,$l$}defined for integers 1≤$l$<$n$and a vector α∈ℝ^{$n$}by letting $$\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ is the generalised cone consisting of all vectors with the height attained among the first$l$coordinates. We show that the exponent takes all values in the interval [$l$+1,∞), with the value$n$attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ_{$n$,$l$}(α)=μ for μ≥$n$. Finally, letting$w$_{$n$}denote the exponent obtained by removing the restrictions on $\underline{x}$ , we show that there are vectors α for which the gaps in the increasing sequence μ_{$n$,1}(α)≤...≤μ_{$n$,$n$-1}(α)≤$w$_{$n$}(α) can be chosen to be arbitrary.