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.