  The parameter reconstruction problem in a sum of Dirac measures from its low
frequency trigonometric moments is well understood in the univariate case and
has a sharp transition of identifiability with respect to the ratio of the
separation distance of the parameters and the order of moments. Towards a
similar statement in the multivariate case, we present an Ingham inequality
which improves the previously best known dimension-dependent constant from
square-root growth to a logarithmic one. Secondly, we refine an argument that
an Ingham inequality implies identifiability in multivariate Prony methods to
the case of commonly used max-degree by a short linear algebra argument,
closely related to a flat extension principle and the stagnation of a
generalized Hilbert function.
