When Do Zeros of an Exponential Sum have Real Parts that Form a Dense Set?

The zeros of exponential sums with complex coefficients and distinct real-valued exponents, also known as exponential polynomials, are among the most intensively studied of all entire functions. Nevertheless, the zeros of exponential sums possess intriguing properties that are yet to be discovered. A prototype worthy of study is the three-term exponential sum in which the zeros are contained in one or at most two disjoint vertical strips in the complex plane. In other words, the real parts of the roots are confined to one or two closed and bounded intervals of the real axis: [a, b] and [c, d]. Here, it is shown that the real parts of the roots are dense in the intervals [a, b] and [c, d] if and only if one key parameter is irrational. This result is an important consequence of Weyl’s equidistribution theorem.