The converse of Bohr's equivalence theorem with Fourier exponents linearly independent over the rational numbers

Given two arbitrary almost periodic functions with Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip V, where both functions assume the same set of values on every open vertical substrip included in V, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be ⁎-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.The first author has been partially supported by a CRM-ISM postdoctoral fellowship and by a fellowship “Ing. Giorgio Schirillo” from INdAM. The second author's research has been partially supported by MICIU of Spain under project number PGC2018-097960-B-C22