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Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. Wesay that $ {\Lambda}$ is type $1$ if the series$s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. Thismeans that for any non-negative measurable $f: {{\mathbb R}}\to [0,+ {\infty})$either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= {{\mathbbR}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f,{\Lambda})=\{x: s(x)=+ {\infty} \}= {{\mathbb R}}$ modulo sets of measure zero.  If $ {\Lambda}$ is not type $1$ we say that $ {\Lambda}$ is type 2.  The exact characterization of type $1$ and type $2$ sets is not known. Inthis paper we continue our study of the properties of type $1$ and $2$ sets. Wediscuss sub and supersets of type $1$ and $2$ sets and we give a complete andsimple characterization of a subclass of dyadic type $1$ sets. We discuss theexistence of type $1$ sets containing infinitely many algebraically independentelements. Finally, we consider unions and Minkowski sums of type $1$ and $2$sets.

Type $1$ and $2$ sets for series of translates of functions