On a Characterization of Shifts of Haar Distributions on Compact Open Subgroups of a Compact Abelian Group

Let X be a compact Abelian group. In the article we obtain a characterization of shifts of Haar distributions on compact open subgroups of the group X by the symmetry of the conditional distribution of one linear form of independent random variables taking values in X given another. Coefficients of the linear forms are topological automorphisms of the group X. This result can be viewed as an analogue for compact Abelian groups of the well-known Heyde theorem, where Gaussian distributions on the real line are characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another.