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

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 $\mathbb{N}$ 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<m_1<m_2<m_3, \end{equation*} where $c_i$ are $S$-units for a suitable set $S$. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at $S$-unit points with Roth’s theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base $g\ge 2$, there exists a constant $q_0(g)$ such that, for every prime $q>q_0(g)$, 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.