Counting solvable 𝒮-unit equations and linear recurrence sequences with zeros

We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a classical idea of P. Erdős (1935). We then use similar ideas to get a tight upper bound on the number of linear recurrence sequences which attain a zero value.