a continuous distribution on the real line, then with probability 1 it is an irrational of type 1. Irrationals that are not algebraic are said to be transcendental, and can have type strictly greater than 1. (However, the transcendental number $e$ is of type 1.) Known upper bounds to the types of $\pi$, $\pi^2$ and $\log2$ are 6.61, 4.45 and 2.58, respectively. Liouville numbers have type $\eta=\infty$. The type of an irrational number is one less than its irrationality measure (or equivalently, one less than its approximation exponent or Liouville-Roth constant). We refer the reader to Ribenboim [@Rib00] for more information about types of irrational numbers.

Next, we introduce notation which helps us to define an approximation to the discontinuous term, an analogue of $d_n(x)$ in (\[eq1.2\]), when $k=2$. (Here, $k$ is as in (\[eq2.1\]).) Assuming that the lattice, on which the distribution of $X_{ji}$ is supported, consists of points $x_j+\nu
e_j$ for integers $\nu$, define $\xi_{jn}=e_j ({\sigma}_j
n_j^{1/2})^{-1}\{
(n_j x_j/e_j)-\lfloor n_j x_j/e_j\rfloor\}$ and $$\label{eq2.8} \xi_n(x)= \bigl\{x-(c_1
\xi_{1n}+c_2 \xi_{2n}) \bigr\} \frac{{\sigma}_1 n_1^{1/2}}{
c_1 e_1},$$ where, recalling that ${\sigma}_j^2={\operatorname{var}}(X_{ji})$, we define $c_j$ for $j=1$ and 2 by $$\label{eq2.9} c_j= \biggl(\frac{n_j^{-1}{\sigma}_j^2}{ n_1^{-1}{\sigma}_1^2+n_2^{-1}{\sigma}_2^2} \biggr)^{ 1/2}.$$

Let ${\alpha}\in(0,{\frac{1}{2}})$ and partition the set of all 