Phase retrieval for probability measures on the group \(\mathbb{Z}_2^3\) (Q6642738)

Phase retrieval on a locally compact group consists of the description of all probability measures on the group, for which the moduli of their characteristic functions are equal to the modulus of the characteristic function of the given measure. In other words, given the moduli characteristic function up to a shift (\(\mu_x(E)=\mu (E+x)\)) and central symmetry, can we retrieve the probability measure? What are the conditions for it to be possible?\N\NFor completion let us recall that two measures are said to be equivalent if the modulus of their characteristic functions is the same. Also, a probability measure \(\mu\) is said to have a trivial equivalence class if the shift \(\mu_x\) and \(\mu^-{}_x\) are equivalent to it.\N\NThe set of probability measures that have a trivial equivalence class is denoted by \(\mathrm{TEC}(X)\). As an example, the normal distribution belongs to \(\mathrm{TEC}(\mathbb{R})\) but the Poisson distribution does not. This paper describes completely the class \(\mathrm{TEC}(\mathbb{Z}^{3}_2)\). To achieve this, the author provides preliminary knowledge on the subject and a brief survey of previous results that can help the reader to better understand the demonstration.