  In this paper, we focus on the following testing problem: assume that we are
given observations of a real-valued signal along the grid $0,1,\ldots,N-1$,
corrupted by white Gaussian noise. We want to distinguish between two
hypotheses: (a) the signal is a nuisance - a linear combination of $d_n$
harmonic oscillations of known frequencies, and (b) signal is the sum of a
nuisance and a linear combination of a given number $d_s$ of harmonic
oscillations with unknown frequencies, and such that the distance (measured in
the uniform norm on the grid) between the signal and the set of nuisances is at
least $\rho>0$. We propose a computationally efficient test for distinguishing
between (a) and (b) and show that its "resolution" (the smallest value of
$\rho$ for which (a) and (b) are distinguished with a given confidence
$1-\alpha$) is $\mathrm{O}(\sqrt{\ln(N/\alpha)/N})$, with the hidden factor
depending solely on $d_n$ and $d_s$ and independent of the frequencies in
question. We show that this resolution, up to a factor which is polynomial in
$d_n,d_s$ and logarithmic in $N$, is the best possible under circumstances. We
further extend the outlined results to the case of nuisances and signals close
to linear combinations of harmonic oscillations, and provide illustrative
numerical results.
