New large value estimates for Dirichlet polynomials

The authors state in Section 1.1 that 'Our argument crucially relies on the stronger ℓ∞ assumption to bound the additive energy E(W) efficiently'. While this is clearly stated, it would be beneficial for the reader if the authors could provide a more detailed discussion on why the standard ℓ2-based approaches fail to capture the necessary cancellation in the cubic trace expansion. A brief comparison with the methods used in [HB] or [Ju] regarding the treatment of the coefficients would help clarify the specific technical barrier that the ℓ∞ assumption helps to bypass.

In the proof of Proposition 3.1, the authors define the set W' to be T^η-separated. The transition from the bound on W' to the bound on W is handled by a subdivision argument: 'If instead T > N^{6/5}, then we divide W' into ⌈T/N^{6/5}⌉ subsets W'j'. It is not immediately obvious that the sum of the bounds for each subset W'j preserves the required precision in the exponent. A more explicit verification that the 'o_η(1)' terms do not accumulate in a way that compromises the final result would strengthen the presentation of this reduction.

The authors state in Section 1.1 that 'Our argument crucially relies on the stronger ℓ∞ assumption to bound the additive energy E(W) efficiently'. While this is clearly stated, it would be beneficial for the authors to elaborate on the specific structural reasons why the ℓ∞ constraint is so much more powerful than the ℓ2 constraint in the context of their specific Fourier-analytic approach. A more explicit discussion of the 'remark after Lemma 11.4' would help readers understand the fundamental limitations of the method.

Regarding the proof of Proposition 9.1, the authors claim that 'the integrand of J(f) is of size M^4 on a set of measure ⪆ B^2/T'. This claim is central to the argument, but the justification provided in the remark is somewhat heuristic. Providing a more formal derivation or pointing to a specific lemma in the literature that justifies this measure-theoretic claim would strengthen the argument significantly. The leap from the heuristic to the formal bound in the proof of Lemma 9.2 is quite large and deserves more explicit treatment.

The authors mention that 'the bound of Theorem 1.1 can be compared with the bound (1.1) ... coming from combining the classical Mean Value Theorem for Dirichlet polynomials and the Montgomery-Halász-Huxley large values estimate'. It would be very helpful if the authors provided a figure or table explicitly comparing the new bounds with the classical ones in the (N, V) plane, as the current text relies on qualitative descriptions like 'stronger bound when N^{7/10+ϵ} < V < N^{8/10-ϵ}'. A visual representation would clarify the scope of the improvement.

The treatment of the S_3 term in Section 7 is the core of the paper's novelty. The authors define the function R(v) and then state that 'the cancellation in the inner integral when |m_1 v_1 + m_2 v_2 + m_3| is not ⪅ 1/N is one of the key observations in our proof'. This is a very elegant observation, but the authors could provide a more detailed example or a simpler case to illustrate why this cancellation is so specific to Dirichlet polynomials, as it is a powerful and non-obvious insight.

The authors state in Section 1.1 that 'Our argument crucially relies on the stronger ℓ∞ assumption to bound the additive energy E(W) efficiently.' While this is clearly stated, it would be beneficial for the reader if the authors could provide a more detailed discussion on why the ℓ2 constraint fails in this specific framework. Specifically, where exactly in the expansion of the trace or the application of Heath-Brown's theorem does the ℓ∞ bound become indispensable? Providing a counterexample or a simple case where the ℓ2 bound leads to a divergence would clarify the necessity of this assumption for the broader community.

In the proof of Proposition 9.1, the authors define J(f) and claim that 'the sets {m1U + m3/m2} m1,m2,m3 ∼ M cannot overlap too much.' This is a powerful heuristic, but the transition to the formal bound in Lemma 9.2 is quite abrupt. Specifically, the introduction of the smoothed function e_f and the subsequent application of the induction hypothesis on ϵ requires a very careful reading. It would be helpful if the authors could provide a brief intuitive explanation of why the smoothing by T-intervals is the 'correct' way to handle the Fourier decay of f, and whether this choice is optimal or if other smoothing kernels would yield similar results.