Spectrality of a class of infinite convolutions with and without compact supports in Rᵈ

Generalizing a result given by Li, Miao and Wang in 2022, we study the spectrality of a class of infinite convolutions in Rᵈ generated by sequences of nearly d-th power lattices. This allows us to easily construct spectral measures with and without compact supports in Rᵈ. According to a result on the relation between supports of infinite convolutions and sets of infinite sums, we systematically study the Hausdorff and packing dimensions of infinite sums of finite sets in Rᵈ. As an application, we give concrete formulae for the Hausdorff and packing dimensions of the supports of a class of spectral measures in Rᵈ with the form of infinite convolutions generated by specific sequences of nearly d-th power lattices, and finally we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in Rᵈ.