A proof of the Erdős primitive set conjecture

The core of the proof rests on the 'key new ingredient (Proposition 3.3)' which bounds the sum of densities by a factor of sqrt(v). The logic here is subtle: the author claims that 'the worst-case scenario is when the subset of a in A with P(a)^(1+v) approx a contributes about d/dv [sqrt(v)] = 1/(2sqrt(v)) to the sum of densities'. While the subsequent integration leads to the correct pi/4 factor, the transition from the discrete sum of densities to the continuous integral requires more explicit justification regarding the convergence of the partition refinement. I suggest the authors add a brief remark or a more formal lemma clarifying the error terms involved in the limit k -> infinity for the partition v_i = i/k.

Regarding the 'L-primitive sets' definition, the author states 'A is L-primitive if no member of A is an L-multiple of another'. This definition is central to the entire paper, yet the distinction between 'primitive' and 'L-primitive' is quite fine. In Section 1.2, the author notes that 'the bound e^gamma is essentially best possible for L-primitive sets... which is markedly larger than f(P)'. It would be beneficial for the reader if the author provided a more explicit example of an L-primitive set that is not primitive to better illustrate why the L-primitive bound is weaker than the primitive bound, as this highlights the 'subtlety of Conjecture 1.1' mentioned in the text.

The author's use of the 'L-primitive' definition, where 'a set A is L-primitive if no member of A is an L-multiple of another', is a clever workaround for the difficulties inherent in the original primitive set definition. However, the manuscript would benefit from a more explicit discussion on the limitations of this definition compared to the standard one. Specifically, when the author states that 'this definition is weaker than primitive', it would be helpful to provide a concrete example of a set that is L-primitive but not primitive to clarify the distinction for the reader.

The proof of Theorem 1.2 hinges on the 'key new ingredient (Proposition 3.3)', which bounds the sum of densities in the critical range. The logic here is elegant, but the transition from the discrete sum to the integral '∫ 1 0 d v/ 2 √ v (1 + v) = π/ 4' is quite rapid. Providing a more detailed step-by-step derivation of this integral, perhaps in an appendix, would greatly improve the accessibility of this crucial part of the argument for specialists who may want to verify the constant factor.

Regarding the 'Erdős–Sárközy–Szemerédi' conjecture (Conjecture 1.4), the author provides a significant step forward with Theorem 1.5. The discussion of why the limit must attain an equality of 1 is well-reasoned, but the connection between the rate of decay 'O_ε(k^{ε-1/2})' in (1.3) and the final bound of 1.399 could be more explicitly linked. Clarifying how the current method might be pushed to reach the limit of 1 would provide a more satisfying conclusion to this section.

The core of the proof rests on the 'L-primitive' definition (Definition 1.6). The author claims that 'A is L-primitive if no member of A is an L-multiple of another,' which is a stronger condition than the standard primitive definition. I found the transition in Section 1.1, where the author states 'the key new ingredient (Proposition 3.3) shows that if P(a)^(1+v) > a uniformly... then we can bound the sum of densities in (1.4) by sqrt(v),' to be the most critical logical leap. The author should provide more explicit detail on how the 'savings of integral 1 to 0 dv/2sqrt(v)(1+v) = pi/4' is derived, as this constant is central to the final bound. A more step-by-step derivation of this integral would greatly improve the accessibility of the argument.

In Section 4, the proof of Theorem 4.4 involves a complex partition of the set A into A_0 and A_k. The author states, 'we compute the constants C1, C2' and then provides values like 'C2 := sum_{p>23} b_p f(p) ... = 0.251135...'. While these constants appear to be calculated based on the provided tables, the manuscript does not explicitly detail the computational method used to reach these specific values. Given the importance of these bounds to the final result (1.595 < 1.6366), it is essential that the author clarifies whether these are rigorous upper bounds obtained via interval arithmetic or numerical approximations, and provides a reference or appendix for the computational verification.