The distribution of totatives

It is clear that Ak and Bk are subsets of Ck, and in fact Lehmer [3] proved that Ak ⊂ Bk ⊂ Ck. It is not difficult to show that Cp ⊂ Ap for a prime p, so that Ap = Bp = Cp. P. J. McCarthy [5] proved that Ak = Bk when k is not squarefree, and he asked if the result could be extended to all composite numbers k. This was done by P. Erdős [2], who proved that the set Bk \ Ak is infinite for every composite k. Erdős also showed that B2p = C2p for an odd prime p, and then proved that Bk = Ck if k = p and k = 2p, with p odd. The notation (1.2) was introduced by N. G. de Bruijn [1] in the review of [2]. The introduction of these sets Ak, Bk, Ck helps to clarify and summarise the above results. De Bruijn also pointed out that parts of the proof in [2] required rectifications which Erdős was to have supplied, and we give these necessary amendments in Section 2. In [2] Erdős also made the following Conjecture. Let p, q be distinct odd primes such that pq ∈ Ak and pq ≡ −1 (mod k). Then pq ∈ Bk. The study of the distribution of totatives often involves the analysis of the condition under which the sum of two fractional parts of real numbers should exceed 1. In particular we found that the conjecture depends on an interesting inequality associated with residue classes. For a fixed modulus k, we write x < y (mod k) to mean that the least non-negative residue congruent to x is less than that congruent to y. Theorem. Let a, b, c be integers which are distinct (mod k) and satisfying