Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

The paper’s Theorem 1.2 asserts that for G = SL2(C)^n and Γ commensurable with Res_{K/Q}SL2(Z[i]), with K totally imaginary and OK a UFD, the set E(F) of points with bounded orbits under any one-parameter Ad-semisimple subgroup is hyperplane-absolute-winning (HAW) on G/Γ. The proof proceeds by: (i) reducing to the diagonal flow, (ii) establishing a Dani-type correspondence identifying bounded-orbit sets with weighted badly approximable complex vectors (Proposition 3.4), (iii) proving that these Diophantine sets are HAW in C^n (Proposition 4.6), relying critically on a UFD-based ‘simplex’ lemma (Lemma 4.4), and (iv) using invariance of HAW under biholomorphisms to globalize and to handle conjugacy (Proposition 5.1), plus a Jordan–Chevalley reduction for general Ad-semisimple flows (Section 6) . The candidate solution mirrors these steps almost verbatim: reduce to diagonal by conjugacy, parametrize unstable horospherical charts (upper triangular unipotents) as C^n, invoke a Dani correspondence via restriction of scalars, prove the weighted K-badly-approximable set is HAW in C^n using a potential/absolute hyperplane game and a number-field simplex lemma (exploiting OK being a UFD), note HAW’s invariance under biholomorphisms, and patch via charts to conclude E(F) is HAW. The key ingredients in the paper—Dani correspondence (Prop. 3.4), HAW for Bad(K,r) using a UFD-driven lemma (Prop. 4.6 with Lemma 4.4), and biholomorphic invariance (Prop. 5.1)—are all explicitly present and align with the model’s outline . Minor issues: (a) the paper’s notation “Res_{K/Q}SL2(Z[i])” versus Θ(SL2(OK)) is a bit loose (the identification is discussed but would benefit from clarification) ; (b) Proposition 4.6 is proved using the hyperplane-potential (HP) game; the (standard) implication HP-winning ⇒ HAW is not stated explicitly in Section 2 for the complex setting, although the proof of HAW appears to rely on that known implication. This is a small exposition gap, not a substantive flaw. Overall, both arguments are correct and essentially the same.