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 $\phi$ via the property of 'Banach density'. These sets related to the totient function are defined as follows: $V:=\phi(\mathbb{N})$ and $N_{i}:=\{N_{i}(m)\colon m\in V \}$ for $i = 1, 2, 3,$ where $N_{1}(m)=\max\{x\in \mathbb{N}\colon \phi(x)\leq m\}$, $N_{2}(m)=\max(\phi^{-1}(m))$ and $N_{3}(m)=\min(\phi^{-1}(m))$ for $m\in V$. Masser and Shiu (Pacific J. Math. 121(2) (1986) 407-426) called the elements of $N_{1}$ as sparsely totient numbers' and constructed an infinite family of these numbers. Here we construct several infinite families of numbers in $N_{2}\setminus N_{1}$ and an infinite family of composite numbers in $N_{3}$. We also study (i) the ratio $\frac{N_{2}(m)}{N_{3}(m)}$ which is linked to the Carmichael's conjecture, namely, $|\phi^{-1}(m)|\geq 2 ~\forall ~ m\in V$, and (ii) arithmetic and geometric progressions in $N_{2}$ and $N_{3}$. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of $\mathbb{N}$, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.

Correction to: Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:84https://doi.org/10.1007/s12044-019-0512-x