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 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 $N_1$ assparsely 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.