We prove that there are only finitely many odd perfect powers in $\mathbb{N}$ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at $S$-unit points (in a suitable $\nu$-adic convergence), Roth’s...