Singular Linear Forms over Global Function Fields

The paper proves, under the stated PID hypothesis on R_ν, that dim_H Sing(m,n) ≤ mn − mn/(m+n) by (i) an explicit Dani correspondence for the local space X = SL_{m+n}(K_ν)/SL_{m+n}(R_ν) (Lemma 2.1), (ii) constructing a function-field Margulis height with optimal contraction, and (iii) a covering argument yielding the stronger bound for δ-escape-on-average sets (Proposition 6.5), from which Theorem 1.1 follows immediately . By contrast, the model solution appeals to a “general” KKLM theorem as if it were directly applicable in the non-Archimedean function-field setting, asserts an explicit dimension drop κ(a) via an incorrect unstable-Jacobian computation, and dismisses the PID assumption as inessential. The paper itself emphasizes that the PID assumption is needed to use the submodularity of covolume required for the construction of the height function and subsequent estimates . Hence, although the model’s final numerical bound matches the paper’s theorem, its derivation relies on unproven generality and contains computational inaccuracies; the paper’s argument is complete and correct in its stated scope.