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Given two arbitrary almost periodic functions with associated Fourierexponents which are linearly independent over the rational numbers, we provethat the existence of a common open vertical strip $V$, where both functionsassume 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 sameregion of almost periodicity and to be $^*$-equivalent or Bohr-equivalent. Thisresult represents the converse of Bohr's equivalence theorem for thisparticular case.

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