Phase retrieval of entire functions and its implications for Gabor phase retrieval

We characterise all pairs of finite order entire functions whose magnitudes agree on two arbitrary lines in the complex plane by means of the Hadamard factorisation theorem. Building on this, we also characterise all pairs of second order entire functions whose magnitudes agree on infinitely many equidistant parallel lines. Furthermore, we show that the magnitude of an entire function on three parallel lines, whose distances are rationally independent, uniquely determines the function up to global phase, and that there exists a first order entire function whose magnitude on the lines $\mathbb{R} + \tfrac{\mathrm{i}}{n} \mathbb{Z}$ does not uniquely determine it up to global phase, for all positive integers $n$. Our results have direct implications for Gabor phase retrieval.