Perfect powers with few digits and $S$-unit coefficients

Source abstract

In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\N$ having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation \begin{equation*} y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}, \quad 0 q_0(g)$, no perfect$q$-th power admits a representation with five non-zero digits in base$g$. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.