<p>The classical Khintchine–Jarník Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine approximation. This paper concerns the Lebesgue measure, Hausdorff measure and Fourier dimensions of sets arising in weighted and multiplicative Diophantine approximation. We provide zero-full laws for determining the Lebesgue measure and Hausdorff measure of the sets under consideration. In particular, the criterion for the weighted setup refines a dimensional result given by Li et al. (Adv Math 470:110248, 2025), while the criteria for the multiplicative setup answer a question raised by Hussain and Simmons (J Number Theory 186:211–225, 2018) and extend beyond it. A crucial observation is that, even in higher dimensions, both setups are more appropriately understood as consequences of the ‘balls-to-rectangles’ mass transference principle. We also determine the Fourier dimensions of these sets. The results we obtain indicate that, in line with the existence results, these sets are generally non-Salem sets, except in the one-dimensional case. This phenomenon can be partly explained by another result of this paper, which states that the Fourier dimension of the product of two sets equals the minimum of their respective Fourier dimensions.</p>