  In this article, we investigate sparse subsets of the natural numbers and
study the sparseness of some sets associated with 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 call the elements of $N_1$ as `sparsely totient
numbers' and construct 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.
