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

## TL;DR

This paper establishes a necessary and sufficient condition for two almost periodic functions with Fourier exponents linearly independent over the rationals to be equivalent, based on their shared value sets in a common vertical strip, thus providing a converse to Bohr's theorem.

## Contribution

It proves a converse to Bohr's equivalence theorem for almost periodic functions with Fourier exponents linearly independent over the rationals, linking shared value sets to equivalence.

## Key findings

- Shared value sets in a vertical strip imply equivalence of functions.
- Functions have the same region of almost periodicity under the given conditions.
- The result characterizes when two functions are $^*$-equivalent or Bohr-equivalent.

## Abstract

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.