Simultaneously bounded and dense orbits for commuting Cartan actions

The paper proves that for X3 = SL3(R)/SL3(Z), if F1 is a regular diagonal flow and F2,j are distinct commuting diagonal flows, then B(F1) ∩ ⋂j D(F2,j+) has full Hausdorff dimension (Theorem 1.2), via entropy-in-the-cusp control, Ledrappier–Young theory, and Marstrand slicing; it explicitly notes it cannot (at present) prove thickness/HAW for this intersection. By contrast, the model’s solution asserts a stronger HAW conclusion for S(F1,F2)=B(F1)∩D(F2+) and for the countable intersection, which is neither stated nor supported by the paper; indeed the authors remark they cannot prove thickness, contradicting the model’s HAW claim. See the statement of Theorem 1.2 and the remark (paper’s Introduction) and the closing step of the proof using slicing for the countable family of flows, as well as the use of high-entropy and non-escape-of-mass arguments (entropy in the cusp) to build the measure-theoretic backbone of the proof.