  We establish a connection between the $L^2$ norm of sums of dilated functions
whose $j$th Fourier coefficients are $\mathcal{O}(j^{-\alpha})$ for some
$\alpha \in (1/2,1)$, and the spectral norms of certain greatest common divisor
(GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain
sharp conditions for the convergence in $L^2$ and for the almost everywhere
convergence of series of dilated functions.
