  Given two arbitrary almost periodic functions with associated 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.
