Two multidimensional analogues of the F. and M. Riesz theorem

A new collection of Riesz sets for $\mathbb R^n$ and $\mathbb Z^n$ is exhibitize. The results are new even in $1$-dimensional case. An asymptotic estimate of Fourier multipliers on finite measures in $H^{1,\infty}$ space used for the proof is interesting in itself.