  We prove that if the system of integer translates of a square integrable
function is $\ell^2$-linear independent then its periodization function is
strictly positive almost everywhere. Indeed we show that the above inference is
true for any square integrable function since the following statement on
Fourier analysis is true: For any (Lebesgue) measurable subset A of [0,1], with
positive measure, there exists a non trivial square summable function, with
support in A, whose partial sums of Fourier series are uniformly bounded.
