Cardinality of subsets of the residue group with nonunit differences of elements

Abstract

Abstract The paper is concerned with the subsets I ⊂ {0, ..., d – 1} for which gcd(n – m, d) ≠ 1 for any n, m ∈ I. Such subsets are called sets of nontrivial differences. Let d > 1 and d 1 be the least prime divisor of d. We prove that the largest cardinality of a set of nontrivial differences is d/d 1. Sets of nontrivial differences in which not all differences of elements are multiples of the same prime factor d are called nonelementary. Let t be the number of prime factors of d. We show that there are no nonelementary sets for t ⩽ 2. It is shown that a minimal nonelementary set may have arbitrary order in the interval 3 , t ¯ $\overline {3,\;t}$ . The largest cardinality of nonelementary sets is estimated from below and above.