AN AVOIDANCE PRINCIPLE AND MARGULIS FUNCTIONS FOR EXPANDING TRANSLATES OF UNIPOTENT ORBITS

The paper’s main theorem (Avoidance Principle) states the precise dichotomy and quantitative bound you summarized, including the uniformity of Tx on compact sets, and provides a complete proof via Margulis functions, an averaging operator At, a recurrence-to-compact estimate for the height h, and a Margulis inequality for auxiliary functions FY attached to closed S-orbits; the proof culminates in the two-step estimate (inj-radius control and avoidance of all small-volume S-orbits) with an R^{-1} bound when D is chosen large, exactly as in Theorem 2 and its proof steps 1–2 (see Theorem 2 and Section 9 with Step 1 and Step 2, including Step 2.1 and 2.2, where counting of closed S-orbits is handled by Theorem 5) . By contrast, the candidate model proposes a Dani–Margulis linearization plus quantitative nondivergence route. While this strategy is plausible in spirit, the writeup crucially assumes two points that are not justified in the paper’s general setting: (i) a geometry-of-numbers–type counting of wedge vectors (or closed S-orbits) with a polynomial bound independent of arithmeticity of Γ, and (ii) finiteness of relevant intermediate subgroups S, which the paper obtains via a separate structural finiteness lemma, not by “finitely many possible dimensions.” The paper explicitly proves the needed counting bound for closed S-orbits (Theorem 5) and uses a finiteness lemma for intermediate subgroups to sum over S, whereas the model’s counting step relies on an implicit Γ-invariant lattice in a representation and on ‘only finitely many dS,’ neither of which suffices in general. Moreover, the paper specifies Tx = h(x)^{1/δF} and uses the Margulis inequality to derive the needed recurrence and Chebyshev bounds, while the model’s claim that one can essentially take TK = 2 on compact K without constructing Tx misses the explicit dependence that the paper establishes. Accordingly, the paper is correct and complete in the claimed generality, whereas the model solution has substantial unproven assumptions in this general setting .