A weak zero-one law for sequences of random distance graphs

We study zero-one laws for properties of random distance graphs. Properties written in a first-order language are considered. For $p(N)$ such that $pN^{\alpha}\to\infty$ as $N\to\infty$, and $(1-\nobreak p)N^{\alpha}\to\infty$ as $N\to\infty$ for any $\alpha>0$, we succeed in refuting the law. In this connection, we consider a weak zero-one $j$-law. For this law, we obtain results for random distance graphs which are similar to the assertions concerning the classical zero-one law for random graphs.
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