Sums of $S$-units in sum of terms of recurrence sequences

Abstract: Let $S := {p_1,\ldots ,p_{\ell}}$ be a finite set of primes and denote by $\mathcal{U}S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n){n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two. In this paper, we provide a finiteness result for the solutions of the Diophantine equation $aU_n + bU_m = z_1 +\cdots +z_r,$ where $n\geq m$ and $z_1, \ldots, z_r\in \mathcal{U}_S$.