Fourier decay of product measures

Can one characterise the Fourier decay of a product measure in terms of the Fourier decay of its marginals? We make inroads on this question by describing the Fourier spectrum of a product measure in terms of the Fourier spectrum of its marginals. The Fourier spectrum is a continuously parametrised family of dimensions living between the Fourier and Hausdorff dimensions and captures more Fourier analytic information than either dimension considered in isolation. We provide several examples and applications, including to Kakeya and Furstenberg sets. In the process we derive a novel Fourier analytic characterisation of the upper box dimension.