A function field analogue of Romanoff's theorem

We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let ϕ be a Drinfeld A-module over a global function field L and denote by ϕ(L) the A-module structure on L coming from ϕ. Let Γ ⊂ ϕ (L) be a free A-submodule of finite rank. For each effective divisor [Formula: see text] of L, let fΓ(𝔇) be the cardinality of the image of the reduction map [Formula: see text] if all elements of Γ are relatively prime to the divisor 𝔇; otherwise, just set fΓ(𝔇) = ∞. We give explicit upper bounds for the series [Formula: see text] and [Formula: see text].