On the number of subsets of the residue ring such that the difference of any pair of elements is not invertible

The paper is concerned with subsets $I$ of the residue group ${Z_d}$ in which the difference of any two elements is not relatively prime to $d$. The class of such subsets is denoted by $U\left( d \right)$, the class of sets from $U\left( d \right)$ of cardinality $r$ is denoted by $U\left( {d,\;r} \right)$. The present paper gives formulas for evaluation or estimation of $\left| {U\left( d \right)} \right|$ and $\left| {U\left( {d,\;r} \right)} \right|$.