  J.C.Lagarias (2000) conjectured that if $\mu$ is a complex measure on
p-dimensional Euclidean space with a uniformly discrete support and its
spectrum (Fourier transform) is also a measure with a uniformly discrete
support, then the support of $\mu$ is a subset of a finite union of shifts of
some full-rank lattice. The conjecture was proved by N.Lev and A.Olevski (2013)
in the case p=1. In the case of an arbitrary p they proved the conjecture only
for positive measures.
  Here we show that Lagarias' conjecture is false in the general case and find
two new special cases when assertion of the conjecture is valid.
