{"id":"909ef975-07e5-4f0b-9fa6-01168610f3b0","arxiv_id":"2501.01634","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"For n up to 186 the paper tabulates exact maximum sizes of 3-free subsets, but its headline large-n comparison uses a sphere variant that is not actually 3-free.","lead":"This paper computes exact sizes of the largest subsets of {1,...,n} with no three evenly spaced numbers for all n up to 186 and compares known construction methods for larger n. A smart generalist might read it to see when asymptotic number theory constructions beat simple tricks in practice, but one central comparison method is not actually valid.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.18 (SPHERE-NZ) is false: a base-9 counterexample shows the union contains a 3-AP, so the Section 5 large-n comparison is unsupported.","rationale":"The reader identified the load-bearing weakness as Theorem 4.18's claim that the SPHERE-NZ union is 3-free, and specifically that Lemma 4.15's necessary conditions are incorrectly used to exclude zero coordinates in the middle vector. My independent check confirms this precisely: the proof of Theorem 4.18 applies condition C to the middle vector y even though y belongs to Ad,s,k, not to Ad,s+1,k,C. The explicit counterexample with d=2, k=2, s=1 in base 9 gives the 3-AP 8, 9, 10, with outer vectors in Ad,s+1,k,C and middle vector in Ad,s,k. Since the Section 5 tables list SP values that match the SPHERE-NZ counts, the main empirical comparison is invalidated. I do not see a separate concern that would change the verdict; the small-n computations and the survey portions are not affected by this specific error, but the central large-n claim fails. Therefore the reader's REJECT verdict remains appropriate and no verdict adjustment is needed.","tokens_in":45116,"tokens_out":5196,"duration_ms":49999,"concrete_test":"For d=2, k=2, s=1, compute Ad,1,2 ∪ Ad,2,2,C in base 9 and enumerate all triples x, y, z in the union with x < y < z and x+z = 2y. The triple of digit vectors (-1,1), (0,1), (1,1) corresponds to the integers 8, 9, 10 and is a valid 3-AP in the claimed set, confirming that Theorem 4.18 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central large-n claim in Section 5 — that for n >= 10^9 the sphere methods beat all others — depends on Theorem 4.18, which asserts that Ad,s,k ∪ Ad,s+1,k,C is 3-free when C is the condition that all coordinates are nonzero. The proof is invalid: in the only unresolved case, Lemma 4.15.6b.ii yields y_i = 0 for the middle vector y, and the proof calls this a contradiction to condition C. But C is imposed only on Ad,s+1,k,C; the middle vector y has norm s and therefore lies in Ad,s,k, where zero coordinates are allowed. The theorem is not merely unproven but false: for d=2, k=2, s=1 in base 9, the vectors (-1,1) and (1,1) have squared norm 2 and all coordinates nonzero, so they lie in Ad,s+1,k,C; the vector (0,1) has squared norm 1, so it lies in Ad,s,k. Their integer values are 8, 9, and 10, which form a 3-AP. Thus Theorem 4.18 is contradicted. Because Table 7 and the SP column of Tables 8-13 use exactly this SPHERE-NZ construction (the 10^9 entry 5.97×10^5 matches the SPHERE-NZ count, not SPHERE-NN), the empirical conclusion that sphere methods produce the largest 3-free sets for n ≥ 10^9 rests on a false construction. The small-n exact values may be correct, but the headline large-n comparison and the claimed superiority of SPHERE-NZ are invalid as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is an empirical survey of methods for constructing large subsets of [n] with no three-term arithmetic progressions (3-free sets). It reports exact values of sz(n) for 1 ≤ n ≤ 186, upper and lower bounds for 187 ≤ n ≤ 250, numerical applications of Roth's theorem to obtain upper bounds, and a comparison of several asymptotic constructions: Base 3, Base 5, KD, Block, and the Sphere methods. The central large-n claim is that, for n ≥ 10^9, the nonconstructive sphere methods, especially the variant called SPHERE-NZ, produce larger 3-free sets than any other tested method.","tokens_in":45443,"tokens_out":4788,"duration_ms":44078,"significance":"If the main claims were valid, the paper would be a useful empirical complement to the asymptotic literature on 3-free sets. The small-n exact values, the careful discussion of backtracking and linear-programming upper bounds, and the numerical comparison of Roth's upper bound against splitting are valuable and appear to be self-contained. The paper also includes reproducible dynamic-programming computations, and the fitted constants in Appendices III and V are descriptive rather than used to derive the central claim. However, the headline comparison for large n rests on Theorem 4.18, which is false. Consequently the claimed superiority of SPHERE-NZ and the associated Section 5 conclusion are unsupported and cannot be repaired by a local correction.","major_comments":[{"comment":"Theorem 4.18 is false. The proof's final step claims that Lemma 4.15.6b.ii, which gives y_i = 0 for some coordinate of the middle vector, contradicts condition C (all coordinates nonzero). But C is imposed only on the sphere of norm s+1, namely A_{d,s+1,k,C}; the middle vector y has norm s and lies in A_{d,s,k}, where zero coordinates are allowed. A concrete counterexample is obtained with d=2, k=2, s=1 in base 9: the vectors x = (-1,1), y = (0,1), z = (1,1) correspond to the integers 8, 9, 10. Both x and z have squared norm 2 with all coordinates nonzero, so they lie in A_{2,2,2,C}; y has squared norm 1, so it lies in A_{2,1,2}. These three integers form a 3-AP, contradicting the theorem.","section":"§4.5.2, Theorem 4.18"},{"comment":"The large-n comparison rests on the invalid SPHERE-NZ construction. Table 7 reports SPHERE-NZ = 5.97×10^5 for n = 10^9, and Table 8's SP column also gives 5.97×10^5 at n = 10^9, even though the text of Section 5 identifies SP as SPHERE-NN. Table 7 gives SPHERE-NN = 3.00×10^5 for n = 10^9, so the SP column is actually using the SPHERE-NZ counts. Because Theorem 4.18 is false, these numbers are not valid lower bounds for sz(n), and the observation that the sphere method beats all other methods for n ≥ 10^9 is unsupported.","section":"§5 and Appendix VII, Tables 7–13"},{"comment":"The empirical conclusion fails even if one substitutes the valid SPHERE-NN variant for the invalid SPHERE-NZ. At n = 10^9, Table 7 gives SPHERE-NN = 3.00×10^5 while Table 8 gives Base 3 = 5.24×10^5, so the valid sphere construction does not beat the Base 3 method at the claimed crossover point. The headline claim that the nonconstructive sphere methods produce the largest sets for n ≥ 10^9 therefore cannot be maintained in the paper's current form.","section":"§5, Observation 1"}],"minor_comments":[{"comment":"There are numerous typographical errors, including \"su ch sets\" in the abstract, \"Erdos\" for \"Erdős\", and \"Health-Brown\" for \"Heath-Brown\" in Section 1.1; these should be corrected.","section":"Throughout"},{"comment":"The notation is inconsistent: the text says the SP column abbreviates SPHERE-NN, but the numerical values correspond to SPHERE-NZ, as noted above. The labels in Tables 8–13 should be reconciled with the definition in Section 4.5.4.","section":"§5"},{"comment":"The table header says \"300,000 < N < 1,000,000\" but the table includes values of N up to 178,970,459; the header or the table should be corrected.","section":"Appendix VIII, Table 16"},{"comment":"The entry for n = 10^11 reports size 1.56×10^7, which is smaller than the neighboring entries for n = 10^10 (2.74×10^7) and n = 10^12 (9.81×10^7). Since the SPHERE method should be monotone in n, this is likely a typo and should be checked.","section":"Appendix V, Table 6"}],"recommendation":"reject","confidential_remarks":"The small-n exact values and the Roth-based upper-bound computations are contributions that could survive in a revised or separate paper, but the central large-n comparison is built on a false theorem. The authors should be given the counterexample in confidence so they can correct or retract the SPHERE-NZ claims before any resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a real contribution in the small-n exact values: sz(n) for n=151..186, new lower bounds for 204, 227, and 233, and a reasonably careful survey of construction methods. The backtracking-with-information and LP-based upper bound techniques are described concretely, and the numerical use of Roth's proof for upper bounds is a nice idea. Those parts deserve credit.\n\nThe soft spot is not minor: Theorem 4.18 is false, and it is load-bearing. The SPHERE-NZ union A_{d,s,k} ∪ A_{d,s+1,k,C} is not 3-free. The proof's final case (a=b=1, c=0) invokes Lemma 4.15.6b.ii to claim the middle vector y has a zero coordinate and calls that a contradiction to condition C. But C only applies to the s+1 sphere; y has norm s and lies in A_{d,s,k}, where zero coordinates are allowed. A concrete counterexample in base 9 with d=2, k=2, s=1: (−1,1) and (1,1) lie in A_{d,2,2,C}, (0,1) lies in A_{d,1,2}, and they map to 8, 9, 10, a 3-AP. Since Section 5 and Appendix VII use the SPHERE-NZ sizes (the n=10^9 entry 5.97e5 matches Table 7), the claim that sphere methods beat everything for n≥10^9 is unsupported. There is also a table inconsistency: the large-n tables label the SP column as SPHERE-NN, but the numbers match SPHERE-NZ.\n\nNo code or certificates are provided, so the exact values to 186 are not independently verifiable from the preprint. The overlap with the authors' earlier paper is disclosed, but it makes the incremental novelty hard to audit.\n\nIf the SPHERE-NZ part were removed and the Section 5 comparison redone using only valid methods, the paper would still be a worthwhile reference for the small-n data and the survey. As it stands, the central empirical conclusion cannot hold. A serious referee should be asked to sort this out, or the authors should retract the invalid construction. I would not cite it in its present form.","headline":"Useful small-n exact values and survey, but the main large-n comparison rests on a false SPHERE-NZ construction.","tokens_in":46044,"tokens_out":2677,"would_cite":false,"duration_ms":25624,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","11B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the nonconstructive sphere method produces the largest known 3-free subsets of {1,...,n} once n reaches 10^9, and reports exact maximum sizes for all n up to 186.","keywords":["arithmetic progressions","3-free sets","sphere method","exact values","empirical crossover","upper bounds","integer programming","Ramsey-type theorems"],"falsifier":"For d=2, k=2, s=1 in base 9, check the three digit vectors (-1,1), (0,1), and (1,1): the outer two have squared norm 2 and no zero coordinates, the middle has squared norm 1, and the corresponding integers form a 3-term arithmetic progression; if all three are included in the SPHERE-NZ union, the claimed 3-freeness fails.","tokens_in":44820,"feed_emoji":"🔢","tokens_out":6822,"duration_ms":68057,"temperature":0.7,"pith_summary":"This paper asks how large a subset of {1,...,n} can be while containing no three equally spaced numbers, a set called 3-free. Its headline empirical claim is that the asymptotically best construction, the nonconstructive sphere method, first overtakes all constructive competitors at about n = $10^{9}$, so from that point on the best known sets come from a method whose proof only shows existence. The paper also pushes exact values of the maximum size sz(n) from n ≤ 150 to n ≤ 186 and gives close upper and lower bounds for 187 ≤ n ≤ 250. These numbers matter because large 3-free sets appear in applications from diagonal queens domination to communication complexity and linearity testing, and the crossover shows that the asymptotic ordering of constructions can govern practical behavior at finite but very large n.","feed_headline":"Sphere method beats all known 3-free-set builders past 10^9","feed_subtitle":"Nonconstructive sphere construction overtakes base-3, base-5, KD, and block methods for n ≥ 10^9, data show.","key_machinery":"The sphere method represents each integer in base 4d+1 with digits in {-d,...,d}, identifies the number with its digit vector, and selects all numbers whose digit vectors have a fixed squared length s. Because x+z=2y would force the digit vectors to satisfy x-vector + z-vector = 2y-vector, and two distinct vectors of length sqrt(s) cannot sum to a vector of length 2sqrt(s), each individual sphere is 3-free. The SPHERE-NZ variant unions the sphere at s with the sphere at s+1 restricted to vectors with no zero coordinate; Theorem 4.18 uses a parallelogram norm identity, Lemma 4.15, to claim the union remains 3-free. The construction is made algorithmic by a dynamic program that counts digit vectors of a given norm, so the optimal parameters (d,s,k) can be found even though the proof of existence is nonconstructive.","core_discovery":"The central discovery is a quantitative crossover among known constructions. When base-3, base-5, KD, block, and sphere methods are all coded and compared for n from $10^{1}$ up to $10^{65}$, the sphere method produces larger 3-free sets than every other method for every n ≥ $10^{9}$. The authors support the sphere construction with a new variant, SPHERE-NZ, which takes the union of two spheres of squared norms s and s+1 and imposes a nonzero-coordinate condition on the outer sphere; Theorem 4.18 asserts that this union is 3-free. In the small-n regime the paper computes sz(n) exactly for 1 ≤ n ≤ 186, gives upper and lower bounds for 187 ≤ n ≤ 250, and obtains new lower bounds such as sz(204) ≥ 42, sz(227) ≥ 45, and sz(233) ≥ 46. It also shows that using the analytic upper-bound proof numerically improves upper bounds on sz(N) by roughly 17 percent for the largest N tested.","pith_inferences":["If the theoretical gap in the proof of Theorem 4.18 is real, the empirical crossover claim would still be testable by verifying the 3-freeness of each constructed set directly, though the validity of the SPHERE-NZ union would then need a corrected condition or proof.","The estimated constants c for sphere and block methods, around 2.55 and 4.3 respectively, suggest the finite-n data already reflect the asymptotic quality ordering, so the crossover point at 10^9 may serve as a practical threshold for when asymptotic results dominate elementary constructions.","The same empirical comparison could be extended to k-APs for k ≥ 4 and to the higher-level sphere unions sketched in Theorem 4.23, where the required conditions grow more complex; the paper leaves that as future work.","Because the paper notes that larger 3-free sets do not automatically improve matrix multiplication algorithms, the practical payoff of the crossover is more direct in communication complexity and linearity testing than in algebraic complexity."],"forward_implications":["For n at least 10^9, the best known 3-free sets come from the nonconstructive sphere method, so anyone needing large 3-free sets in practice can use the optimized search over (d,s,k) despite the absence of an explicit construction.","The exact values sz(n) for n ≤ 186, the bounds for 187 ≤ n ≤ 250, and the new lower bounds for n = 204, 227, and 233 update the small-n record.","Among the sphere variants tested, SPHERE-NZ is the best for most n, beating plain SPHERE and SPHERE-NN in the authors' tables.","Using the analytic upper-bound proof numerically gives upper bounds on sz(N) that are about 17 percent better than elementary splitting for the largest tested N.","Beyond n ≈ 10^90, the block method overtakes the KD method, and the empirical ordering of all methods appears to settle into the asymptotic ordering."],"supporting_citations":[{"why":"Supplies the sphere construction and the nonconstructive lower bound that the paper's sphere method optimizes.","marker":"[3]"},{"why":"Gives the constructive block method, one of the main competitors in the empirical comparison.","marker":"[21]"},{"why":"Gives the KD method, the first disproof of the conjecture that sz(n) is essentially n^{log_3 2}, and a competitor in the tables.","marker":"[25]"},{"why":"Provides the base-3 method and the earliest small-n computations of sz(n).","marker":"[9]"},{"why":"Supplies the analytic upper-bound proof that Section 6 turns into numerical upper bounds on sz(N).","marker":"[23]"},{"why":"Provides prior exact values of sz(n) and the nonaveraging-set notation used to state the new small-n results.","marker":"[34]"},{"why":"The companion paper covering the small-n case, which the present paper overlaps and extends.","marker":"[11]"}],"fun_headline_variants":["Sphere method wins all 3-free-set contests for n ≥ 10^9","New crossover: sphere construction leads all 3-free methods past 10^9","Sphere method outperforms all known 3-free constructions for n ≥ 10^9","Empirical survey finds sphere method dominates at n ≥ 10^9"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the SPHERE-NZ union is 3-free assumes that every would-be arithmetic progression forces the middle vector to have a zero coordinate where the nonzero condition applies, even though that condition is only imposed on the outer sphere's vectors.","fun_headline_variants_meta":{"raw":{"variants":["Sphere method wins all 3-free-set contests for n ≥ 10^9","New crossover: sphere construction leads all 3-free methods past 10^9","Sphere method outperforms all known 3-free constructions for n ≥ 10^9","Empirical survey finds sphere method dominates at n ≥ 10^9"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1464,"prompt_tokens":920,"completion_tokens":544,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":458}},"tokens_in":536,"tokens_out":544,"duration_ms":4995,"temperature":1.0,"reasoning_tokens":458,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:24:34.606007+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For d=2, k=2, s=1 in base 9, check the three digit vectors (-1,1), (0,1), and (1,1): the outer two have squared norm 2 and no zero coordinates, the middle has squared norm 1, and the corresponding integers form a 3-term arithmetic progression; if all three are included in the SPHERE-NZ union, the claimed 3-freeness fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sphere construction and the nonconstructive lower bound that the paper's sphere method optimizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the constructive block method, one of the main competitors in the empirical comparison."},{"cited_title":"Salem and D","cited_arxiv_id":null,"evidence_quote":"Gives the KD method, the first disproof of the conjecture that sz(n) is essentially n^{log_3 2}, and a competitor in the tables."},{"cited_title":"Erd˝ os and P","cited_arxiv_id":null,"evidence_quote":"Provides the base-3 method and the earliest small-n computations of sz(n)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic upper-bound proof that Section 6 turns into numerical upper bounds on sz(N)."},{"cited_title":"Wroblewski","cited_arxiv_id":null,"evidence_quote":"Provides prior exact values of sz(n) and the nonaveraging-set notation used to state the new small-n results."},{"cited_title":"Gasarch, J","cited_arxiv_id":null,"evidence_quote":"The companion paper covering the small-n case, which the present paper overlaps and extends."}],"review_version":1}