{"id":"174046cd-7dc8-4703-a086-38a64ef61076","arxiv_id":"2508.02433","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A set of primes with lower relative density 5/8 and upper density 1 is constructed that fails to represent infinitely many odd integers as sums of three of its elements, showing Shao's 5/8 threshold is sharp.","lead":"Two open problems of András Sárközy about sums of residue classes and sums of primes are resolved. The paper proves that a density threshold of 5/8 for subsets of primes is the best possible for ternary Goldbach type representations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Selberg-sieve estimate in equation (5) is classical and uniform, and the double-exponential construction satisfies all required inequalities.","rationale":"The reader's weakest assumption is equation (5), the uniform Selberg estimate. I agree that this is the main external input, but after checking the application it is sound: the estimate is classical, valid for all even N, and the restriction to W_{2k} only decreases the count. The remainder of the proof relies on the extreme separation of scales: x_{2k+1} > e^{e^{x_{2k}}} ensures both that x_{2k} <= log log x_{2k+1}, making the deletion bound (6) negligible, and that D = x_{2k+1}/sqrt(log x_{2k+1}) is much larger than x_{2k}, which is needed in Case II of Fact 2 and in the density calculation. The arithmetic of the residue classes A1 = {1,2,4,7,13} mod 15 is correctly used: the sumset A1+A1+A1 misses 14 mod 15. No circularity, missing case, or unproven identity was found. The proof is self-contained apart from standard theorems, and the central claim that Shao's 5/8 threshold is sharp is well supported. Therefore the reader's ACCEPT verdict should remain unchanged, though I note the flagged Selberg estimate is indeed the most external input and worth a targeted verification.","tokens_in":6673,"tokens_out":26065,"duration_ms":258699,"concrete_test":"Independently re-derive equation (5) from Nathanson's Theorem 7.2, tracking the implied constant explicitly; then, for a few blocks k, sample primes p in (x_1, x_{2k}] and compare the true number of representations of x_{2k+1}-p as a sum of two primes with the bound (x_{2k+1}/log^2 x_{2k+1}) prod_{p'|(x_{2k+1}-p)} (1+1/p'), including worst-case N with many small prime factors. If the bound fails for any sampled p, the density preservation in Fact 1 would need re-examination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the construction and the reader's flagged assumption. Equation (5) is the standard Selberg upper bound for the number of representations of an even integer as a sum of two primes, and it applies here with an absolute constant because Q_{2k} is a subset of W_{2k}, which is a subset of the primes, so t_p is bounded by the full Goldbach representation count. The product over prime divisors p' of (x_{2k+1}-p) of (1+1/p') is at most O(log log x_{2k+1}) uniformly in N, and the implied constant in (5) is absolute. The other steps are also sound: the choice x_{2k+1} > e^{e^{x_{2k}}} gives x_{2k} <= log log x_{2k+1}, which justifies the second inequality in (6); equation (7) follows because any representation of x_{2k+1}-p as a sum of two elements of W_{2k} must have both summands exceeding x_{2k+1}/sqrt(log x_{2k+1}); and the density preservation Q(x) ~ W(x) uses pi(x_{2k+1}/sqrt(log x_{2k+1})) >> x_{2k+1}/(log x_{2k+1})^{3/2} to dominate the o(x_{2k+1}/log x_{2k+1}) deleted primes. The case analysis in Fact 2 is correct. I find no load-bearing flaw; the central claim appears valid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses two problems from Sárközy's 2001 collection. For Problem 66, it gives explicit counterexamples showing that the answer is negative for both parts: for part (b), a construction modulo 30p with density 5/8 of the reduced residue classes fails to cover a positive proportion of odd classes, and for part (a), a construction modulo 12p and a self-similar construction modulo 30·2^k fail to cover all even classes. It also quotes Shao's positive result showing that 5/8 is sharp for odd squarefree moduli. The main result, Theorem 2, concerns Problem 67: there exists an infinite set Q of primes with upper relative prime density 1 and lower relative prime density 5/8 such that infinitely many odd integers are not representable as a sum of three elements of Q. This shows that Shao's density threshold 5/8 in Proposition 2 is best possible. The proof constructs Q in alternating intervals: in odd-indexed intervals all primes are taken, in even-indexed intervals only primes in five residue classes modulo 15 (density 5/8), then removes a thin top interval from even blocks (Step 2) and removes all primes that appear in representations of x_{2k+1}-p as sums of two primes from the even block (Step 3). A Selberg sieve estimate bounds the number of removed primes, and a case analysis shows that x_{2k+1} itself is not in Q+Q+Q.","tokens_in":7005,"tokens_out":6908,"duration_ms":68373,"significance":"If correct, Theorem 2 is a substantial and somewhat surprising result: it shows that Shao's density theorem has the sharp threshold 5/8, in the sense that a set of primes with lower relative density exactly 5/8 can fail the ternary representation property. This complements the earlier counterexample of Yang and Togbé, whose lower density was only 1/3. The paper also contributes clean counterexamples for Problem 66 and clarifies the statement of the problem by noting that 'reduced' is necessary. The proof is elementary in structure and relies on a classical Selberg sieve estimate, which is a standard and robust tool. The main strengths are the explicit recursive construction, the uniform sieve bound, and the careful case analysis in Fact 2. The result is likely to be of interest to researchers in additive number theory and prime density problems.","major_comments":[{"comment":"The claim that the alternating construction gives d(H)=1 and d(H)=5/8 is plausible but is stated without proof. Since the intervals grow double-exponentially, each new interval contributes a density close to either 1 or 5/8, and previous intervals become negligible; however, the limiting argument should be made explicit, especially because the lower density is defined by a liminf over all x and one must verify that no intermediate value of x dips below 5/8.","section":"Section 3, Step 1, definition of H"},{"comment":"The Selberg sieve estimate in (5) is the main external input, and its uniform applicability should be justified more carefully: the number N=x_{2k+1}-p varies with p, and the product over prime divisors is bounded by O(log log N) only if one invokes the standard estimate σ(rad(N))/rad(N) ≪ log log N. The authors should state explicitly that this bound is uniform for all N in the interval [x_{2k+1}-x_{2k}, x_{2k+1}] and that the implied constants in (5) are absolute; as written, the transition from the first inequality to the second is a bit terse.","section":"Section 3, Step 3, equation (5)"},{"comment":"The proof of Q(x)∼W(x) should be written out for arbitrary x, not just for x near x_{2k+1}. In particular, one should note that all deleted primes in the k-th even block lie above x_{2k+1}/√log x_{2k+1}, and that the total number of primes deleted in all previous blocks is o(x/log x) for x in the current block; this is true because x_{2k+1}/√log x_{2k+1} vastly exceeds x_{2k}, but the argument deserves a sentence to avoid leaving the impression that only the locally deleted primes are controlled.","section":"Section 3, Fact 1, density preservation"}],"minor_comments":[{"comment":"The word 'reduced' is missing in the statement of Problem 66(a) and (b); the authors note this at the start of Section 2 and provide a counterexample for non-reduced sets, but the main text of Section 1 should perhaps flag this immediately, since the problem as quoted is literally false.","section":"Abstract and Section 1, Problem 66 statement"},{"comment":"In the construction A={30k+1,30k+7,30k+13,30k+17,30k+19: 1≤30k≤m}, the notation is slightly ambiguous: the variable k should be an integer, and the condition should probably be 0≤30k<m or 1≤30k≤m with a clear convention. Also, the assertion |A|≥m/6−5 should be derived explicitly: there are 5 residue classes, each contributing about m/30 elements, with at most one multiple of p removed per class.","section":"Section 2, construction modulo 30p"},{"comment":"The sentence 'Similar discussions as above lead to |A|≥2φ(m)/3 as well as that no element of {12k+4 (mod m):1≤12k≤m} is in A+A' should be rephrased for clarity; it is missing a verb and the reader must infer that the omitted element classes are exactly the even classes congruent to 4 mod 12.","section":"Section 2, modulo 12 construction"},{"comment":"In equation (2), the interval [x_{2k+1}-x_{2k}-x_{2k+1}/√log x_{2k+1}, x_{2k+1}] is written as a set difference for W_{2k}; it would be clearer to state explicitly that W_{2k} consists of primes in (x_{2k}, x_{2k+1}] that are not in this interval, since the interval may not be integer-aligned.","section":"Section 3, Step 2, interval notation"},{"comment":"The statement 'By (3) and (4), x_{2k+1}-q_1 ≠ q_2+q_3' is correct only because q_1 runs over primes p with x_1<p≤x_{2k}; the reader should be reminded that q_1 could be ≤x_1, but q_1 is assumed not in Q_{2k} and q_1≤x_{2k} (since q_1≤q_2≤q_3 and q_1+q_2+q_3=x_{2k+1}), so q_1 is indeed a p in that range. This is implicit in the construction and should be stated.","section":"Section 3, Fact 2, Case IV"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well within the scope of the journal and the central result is sound. The proof relies on a standard sieve estimate, and the main technical claims are verifiable. I have no concerns about novelty or attribution: the paper correctly credits Shao, Shen, and Yang–Togbé, and the construction is genuinely new. The remaining issues are presentation-level: several density and uniformity arguments are stated too tersely and should be expanded for the reader's convenience, but none of them affects the validity of the conclusion. I would recommend accepting after minor revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper proves the threshold in Shao's density version of ternary Goldbach is sharp. The authors construct a set Q of primes with upper relative density 1, lower relative density 5/8, and infinitely many odd integers that are not sums of three primes from Q. That directly matches Shao's Proposition 2, so the 5/8 boundary is exactly right.\n\nThe main theorem is a genuine new result. Yang and Togbé had a similar construction with lower density 1/3; the double-exponential block construction here pushes it to 5/8. The even-modulus counterexamples to Sárközy's Problem 66(a) are also new, and the dyadic recursion is tidy. The paper gives credit where due: Shao's residue set mod 15 is the engine, and the Selberg upper-bound sieve is the only external analytic input. I checked the stress-test concern about equation (5); it is the standard uniform bound on Goldbach representations, and since Q2k is a subset of W2k, the restricted count inherits the upper bound. No problem there.\n\nThe soft spots are minor. Fact 1's density preservation argument is compressed; it works because the removed elements lie above x_{2k+1}/sqrt(log x_{2k+1}) and their total count is o(x_{2k+1}/log x_{2k+1}), so Q(x) ~ W(x). But the reader has to fill in the details. In Fact 2, the case split is correct but the exhaustiveness is implicit: because q1<=q2<=q3, any element outside Q2k is from an earlier block and hence <= x_{2k}, so the only possible patterns are the ones listed. The text could say this in one sentence. There are a few typos (\"squre-free\", \"predicition\"), and the abstract's \"sparse subset\" is the wrong word for a set with density 5/8. The note about Helfgott's final version is stale but harmless.\n\nAll told, the mathematics holds up. The construction is explicit, the estimates are standard, and the conclusion is sharp. It's a good short paper for additive number theorists. I would send it to a competent referee and expect acceptance after minor revision. Definitely worth a cite.","headline":"A sharp 5/8 counterexample for the density three-prime problem; the proof is sound and the result is worth publishing.","tokens_in":7497,"tokens_out":9396,"would_cite":true,"duration_ms":92829,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11A05","11P32"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper settles two open problems: it gives counterexamples to both parts of the residue-class sumset problem, and it constructs a prime set of lower relative density 5/8 that leaves infinitely many odd integers unreachable as…","keywords":["three-prime sum problems","sumsets","primes in arithmetic progressions","congruence obstacle","residue classes","density version of the three-prime theorem","relative prime density","sparse subsets of primes"],"falsifier":"Choose a moderate starting size and run the block construction for the first few blocks; the claim predicts that each trap integer $x_{2k+1}\\equiv 29 \\pmod{30}$ has no three-prime representation from $Q$, and that the number of deleted primes is tiny compared with the block size. Finding a representation, or finding deletion counts comparable to the block size, would refute the proof.","tokens_in":6510,"feed_emoji":"➕","tokens_out":19519,"duration_ms":200426,"temperature":0.7,"pith_summary":"The paper answers two open problems from a collection of unsolved problems in number theory. For Problem 66, about sumsets of reduced residue classes modulo $m$, it shows that both parts fail as stated: there exist sets $A$ with $|A| > (\\tfrac{1}{2}+\\varepsilon)\\varphi(m)$ whose two-fold or three-fold sumset misses a positive proportion of even or odd residue classes. It also records that if the hypothesis is raised to $|A| > \\tfrac{5}{8}\\varphi(m)$ and $m$ is odd and square-free, the ternary statement becomes true, so the corrected version is sharp. For Problem 67, the main theorem constructs an infinite set of primes $Q$ with upper relative density $1$ and lower relative density exactly $\\tfrac{5}{8}$ such that infinitely many odd integers cannot be written as a sum of three elements of $Q$. This proves that the $\\tfrac{5}{8}$ threshold in the density version of the three-prime theorem is best possible.","feed_headline":"5/8 prime density can still fail three-prime sums","feed_subtitle":"Shows the 5/8 threshold in the density three-prime theorem is sharp.","key_machinery":"The construction alternates two types of prime blocks: odd-indexed blocks contain all primes in intervals $(x_{2k-1},x_{2k}]$, which pushes the upper relative density to $1$, and even-indexed blocks contain only primes congruent to an element of $A_1=\\{1,2,4,7,13\\}$ modulo $15$, which contributes the lower density $\\tfrac{5}{8}$ via the standard count of primes in arithmetic progressions. The load-bearing estimate is a sieve upper bound, equation (5), of the form $t_p \\ll x_{2k+1}(\\log\\log x_{2k+1})/(\\log x_{2k+1})^2$ for the number $t_p$ of representations of $x_{2k+1}-p$ as a sum of two primes; it ensures that only $O(x_{2k+1}(\\log\\log x_{2k+1})^2/(\\log x_{2k+1})^2)$ primes are deleted. The 'trap' integers are the $x_{2k+1}$, chosen odd and $\\equiv 29 \\pmod{30}$; since no sum of three elements of $A_1$ is $14 \\pmod{15}$, three primes from an even block can never reach them, and the deletion rule eliminates the mixed cases.","core_discovery":"The central discovery is that the $\\tfrac{5}{8}$ density threshold for the three-prime representation theorem is best possible: there exists an infinite set of primes $Q$ with lower relative density exactly $\\tfrac{5}{8}$ and upper relative density $1$ for which infinitely many odd integers $2n+1$ are not sums of three elements of $Q$. The proof interleaves blocks of all primes with blocks of primes in the five residue classes $1,2,4,7,13 \\pmod{15}$, then deletes from each even block every prime that would take part in a representation of a chosen bad integer $x_{2k+1}\\equiv 29 \\pmod{30}$. A standard sieve bound, quoted as equation (5), keeps the deleted set sparse enough that the densities are unchanged; the residue-class structure and the deletion rule then block every possible three-prime representation. The paper also answers Problem 66 negatively in both parts with explicit counterexamples modulo $12p$ and $30p$.","pith_inferences":["Deleting additional sparse primes from the constructed $Q$ preserves the infinitely many unrepresentable odd integers while lowering the lower relative density arbitrarily below $\\tfrac{5}{8}$, so the failure phenomenon is not confined to the exact threshold.","The recursive definition sets $x_{2k+2}=e^{e^{x_{2k+1}}}$, so the unrepresentable integers guaranteed by the theorem are of tower-of-exponential size; effective versions would require explicit constants in the sieve bound.","A natural testable extension is whether failure can be forced for almost all odd integers, rather than infinitely many, at some fixed lower density, and what the maximal exceptional set can be at density $\\tfrac{5}{8}$.","The alternating-block mechanism should transfer to other small moduli: any subset of reduced residue classes whose ternary sumset misses a class would yield a similar sharpness example if the analogous representation bound holds."],"forward_implications":["The strict inequality in the density three-prime theorem cannot be weakened to equality: at lower density exactly $\\tfrac{5}{8}$, the conclusion can fail.","Problem 66 is fully resolved: both parts are false as originally stated, while the ternary statement becomes true and sharp when the threshold is $\\tfrac{5}{8}$ and $m$ is odd and square-free.","A single residue-class obstruction, $14 \\pmod{15}$, can force infinitely many odd integers to be unrepresentable even when the prime set has upper density $1$.","The counterexample sets have positive lower density, so the failure is not a consequence of sparseness but of how the missing primes are distributed."],"supporting_citations":[{"why":"Source of Problems 66 and 67; frames the questions that the paper resolves.","marker":"[16]"},{"why":"Supplies the 5/8 density theorem (Proposition 2) whose sharpness the paper proves, and the residue set A1={1,2,4,7,13} mod 15 whose ternary sumset misses class 14 mod 15.","marker":"[17]"},{"why":"Supplies the sieve estimate (5) bounding the number of representations of x_{2k+1}-p as a sum of two primes, keeping the deleted primes sparse.","marker":"[12]"},{"why":"Supplies the count of primes in arithmetic progressions used to compute the relative densities 1 and 5/8 of the alternating blocks.","marker":"[5]"},{"why":"Earlier counterexample to Problem 67 with lower density 1/3; the present construction improves the lower density of a counterexample set to 5/8.","marker":"[20]"}],"fun_headline_variants":["5/8 density threshold sharp for three-prime sums","Three-prime sums can fail at 5/8 density","Prime set at 5/8 density blocks triple sums","Best possible bound: 5/8 for prime triple sums","Density 5/8 not enough for three-prime sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on a known estimate that each number $x_{2k+1}-p$ has very few splittings into a sum of two primes, uniformly for every prime $p$ up to $x_{2k}$ with one constant that works for all blocks; if that uniformity failed, the deletion step could remove too many primes and the set's lower density would fall below $\\tfrac{5}{8}$.","fun_headline_variants_meta":{"raw":{"variants":["5/8 density threshold sharp for three-prime sums","Three-prime sums can fail at 5/8 density","Prime set at 5/8 density blocks triple sums","Best possible bound: 5/8 for prime triple sums","Density 5/8 not enough for three-prime sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000662,"raw_usage":{"total_tokens":2945,"prompt_tokens":786,"completion_tokens":2159,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":402,"completion_tokens_details":{"reasoning_tokens":2088}},"tokens_in":402,"tokens_out":2159,"duration_ms":20399,"temperature":1.0,"reasoning_tokens":2088,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:58:10.090579+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a moderate starting size and run the block construction for the first few blocks; the claim predicts that each trap integer $x_{2k+1}\\equiv 29 \\pmod{30}$ has no three-prime representation from $Q$, and that the number of deleted primes is tiny compared with the block size. Finding a representation, or finding deletion counts comparable to the block size, would refute the proof.","supporting_citations":[{"cited_title":"S´ ark¨ ozy, Unsolved problems in number theory","cited_arxiv_id":null,"evidence_quote":"Source of Problems 66 and 67; frames the questions that the paper resolves."},{"cited_title":"Shao, A density version of the Vinogradov three primes theorem","cited_arxiv_id":null,"evidence_quote":"Supplies the 5/8 density theorem (Proposition 2) whose sharpness the paper proves, and the residue set A1={1,2,4,7,13} mod 15 whose ternary sumset misses class 14 mod 15."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sieve estimate (5) bounding the number of representations of x_{2k+1}-p as a sum of two primes, keeping the deleted primes sparse."},{"cited_title":"Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74 (Springer, New York, 1980)","cited_arxiv_id":null,"evidence_quote":"Supplies the count of primes in arithmetic progressions used to compute the relative densities 1 and 5/8 of the alternating blocks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier counterexample to Problem 67 with lower density 1/3; the present construction improves the lower density of a counterexample set to 5/8."}],"review_version":1}