DEGREE LOWERING FOR ERGODIC AVERAGES ALONG ARITHMETIC PROGRESSIONS

The paper’s Theorem 1.11 shows that if (i) some GHK seminorm controls the averages and (ii) degree d+1 control implies degree d control (for Zd-measurable functions), then the averages are controlled by degree d; this is implemented via a degree-lowering scheme (Propositions 5.1–5.3) and standard monotonicity of GHK seminorms. The candidate solution abstracts the same skeleton: (1) assume existence of P(s0), (2) use monotonicity to lift to any higher index, and (3) iteratively apply degree reduction P(r+1) ⇒ P(r) down to d. The only caveat is that the candidate treats the step-lowering P(r+1) ⇒ P(r) for all r ≥ d as an assumption, whereas the paper proves the needed step-lowering (from d+1 to d, then bootstraps) under its hypotheses. Aside from this, the logic and conclusion match the paper.