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We say that $ {\\Lambda}$ is type $2$ if the series $s(x)=\\sum_{\\lambda\\in\\Lambda}f(x+\\lambda)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable \"witness function\"\n  $f: {\\mathbb R}\\to [0,+ {\\infty})$ such that both the convergence set $C(f, {\\Lambda})=\\{x: s(x)<+ {\\infty} \\}$ and its complement the divergence set $D(f, {\\Lambda})=\\{x: s(x)=+ {\\infty} \\}$ are of positive Lebesgue measure. 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