Sparse subsets of the natural numbers and Euler’s totient function

Abstract

In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated to the Euler’s totient function φ via the property of ‘Banach density’. These sets related to the totient function are defined as follows: V := φ(N) and Ni := {Ni (m) : m ∈ V} for i = 1, 2, 3, where N1(m) = max{x ∈ N: φ(x) ≤ m}, N2(m) = max(φ −1(m)) and N3(m) = min(φ −1(m)) for m ∈ V. Masser and Shiu (Pacific J. Math. 121(2) (1986) 407–426) called the elements of N1 as ‘sparsely totient numbers’ and constructed an infinite family of these numbers. Here we construct several infinite families of numbers in N2 \ N1 and an infinite family of composite numbers in N3. We also study (i) the ratio N2(m) N3(m) which is linked to the Carmichael’s conjecture, namely, |φ −1(m)| ≥ 2 for all m ∈ V, and (ii) arithmetic and geometric progressions in N2 and N3. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of N, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.