  We consider the problem of estimating the unknown response function in the
multichannel deconvolution model with a boxcar-like kernel which is of
particular interest in signal processing. It is known that, when the number of
channels is finite, the precision of reconstruction of the response function
increases as the number of channels $M$ grow (even when the total number of
observations $n$ for all channels $M$ remains constant) and this requires that
the parameter of the channels form a Badly Approximable $M$-tuple.
  Recent advances in data collection and recording techniques made it of urgent
interest to study the case when the number of channels $M=M_n$ grow with the
total number of observations $n$. However, in real-life situations, the number
of channels $M = M_n$ usually refers to the number of physical devices and,
consequently, may grow to infinity only at a slow rate as $n \rightarrow
\infty$. When $M=M_n$ grows slowly as $n$ increases, we develop a procedure for
the construction of a Badly Approximable $M$-tuple on a specified interval, of
a non-asymptotic length, together with a lower bound associated with this
$M$-tuple, which explicitly shows its dependence on $M$ as $M$ is growing. This
result is further used for the evaluation of the $L^2$-risk of the suggested
adaptive wavelet thresholding estimator of the unknown response function and,
furthermore, for the choice of the optimal number of channels $M$ which
minimizes the $L^2$-risk.
