On a problem of B. Mityagin (Q526861)

With regard to an uncertainty principle, \textit{B. S. Mityagin} [J. Approx. Theory 215, 163--172 (2017; Zbl 1384.46022)] formulated a problem and under certain conditions characterized certain non-empty subsets for which inequalities, given by relations (1) and (2) hold for measurable functions, and further conjectured a concept that is stated as Proposition 1 in the present paper. The proof of this conjecture is given in the paper and it is shown that the claim of the truth of it for \(L^p(\mathbb{R}^d)\) is also true for \(L^\infty(\mathbb{R}^d)\). The condition for this proposition to be sufficient, follows from the method given in [loc. cit.] of the present paper. It is observed, as well claimed by the author of the paper, that the proof given here is abridged and has a different path, without destroying the notion of the conjecture. The necessity part is proved for two distinguished cases, one by considering an additive group that is generated by a non-empty subset \(A\) with vector addition in \(\mathbb{R}^d\) as the group operation, whereas the second case is by considering the additive group to be discrete (briefly speaking, the two cases are for the additive group to be discrete and non-discrete). The completion of the proof of the Proposition 1 (the conjecture) depends on a lemma, which is stated in two parts, is proved in Section 4 of the paper.