Almost perfect inhomogeneous powers in arithmetic progression

Abstract

Let $S$ be a finite set of primes and write $\mathbb{Z}_S$ for the set of those non-zero integers whose prime divisors belong to $S$. Hajdu proved that the abc conjecture implies that the number of terms of any arithmetic progression in $H_S=\{ηx^l\mid η\in \mathbb{Z}_S, x,l\in \mathbb{Z},\ \textnormal{with}\ x>0 \ \textnormal{and} \ l\geq 2 \}$ is bounded. Moreover, if $k\geq 3$ and the exponents of the powers are all $\geq 4$, then the number of such progressions are finite. We consider other sets and prove similar statements for these sets.