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Let $a_i,b_i$ be $n(\geq 2)$ non-zero real numbers. Assuming that $\sum_{i=1}^n a_ie^{\sqrt{-1}b_i x}=1$ has infinite real solutions for $x$, prove or disprove that $b_i(1\leq i\leq n)$ is linearly dependent over the rational numbers $\mathbb Q$. $n=2$ is true by easy computation.

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    $\begingroup$ Where does this come from? $\endgroup$
    – Igor Rivin
    Commented Jul 2, 2018 at 15:34
  • $\begingroup$ When I dealt with an integration by discretization, I got this problem. Although it's not relevant to the original question, I'm curious about this. Thank you for the answer! $\endgroup$
    – Feng Wang
    Commented Jul 6, 2018 at 7:01

1 Answer 1

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This is true and follows from a theorem of G. Polya. See the very nice survey by Langer:

Langer, R. E., On the zeros of exponential sums and integrals., Bulletin A. M. S. 37, 213-239 (1931). ZBL57.0416.04.

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