Intersections in subvarieties of {\mathbb{G}}_{\mathrm{m}}^l and applications to lacunary polynomials

We investigate intersections of a given subvariety X of {\mathbb {G}}_{\mathrm {m}}^l with cosets of 1-parameter subtori, on interpreting the context in terms of S-unit points over function fields. On adopting a function field version of a method introduced recently by the second author, extending to arbitrary dimensions previous work of the first and third authors, we prove that when the number of intersections is substantially higher than expected, one can classify the relevant subtori. As a consequence, we obtain a classification of the cosets of subtori such that there are many multiple intersections with X. This also allows a new proof of a conjecture of Erdős and Rényi on lacunary polynomials. We finally show how the methods yield results in the realm of Unlikely Intersections in {\mathbb {G}}_{\mathrm {m}}^l, and in the last section, reinterpret some of the results in terms of Vojta’s conjecture with truncated counting functions.