{"id":"27c79513-3761-44a1-9e67-b0cf09ca6863","arxiv_id":"2501.11736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each fixed g, η_g(n)/√n converges to a positive finite limit, and α_g(n) = (1 + o_g(1))√(gn).","lead":"This paper proves that the minimal size of a set whose differences cover every number up to n at least g times, divided by sqrt(n), converges to a finite positive limit, answering a question from Kravitz. It also finds the exact asymptotic size of the largest such set with at most g repetitions per difference, a problem tied to coding theory and cryptography.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4 as stated is false: Singer supplies q+1 elements but F uses only q, so most nonzero residues are unrepresentable and Theorem 1's stepping argument is unsupported.","rationale":"I read the paper in good faith and agree with the conditional verdict, but the more load-bearing written gap is Lemma 4, not Proposition 2. The paper's F literally uses q of the q+1 Singer elements, making the lemma false as stated, and Corollary 3 and Theorem 1 rest on it. The reader's Proposition 2 concern is also real, but it is a small endpoint adjustment: replacing sqrt(gn)+1 by sqrt(gn+1) in the prime interval restores the claimed N <= n. Both flaws are local and repairable, and the central theorems are likely correct, so I would keep the reader's CONDITIONAL verdict rather than move to accept or reject. My agreement is partial because the main concern I identify differs from the reader's weakest assumption, though the overall assessment is the same.","tokens_in":12352,"tokens_out":16439,"duration_ms":174501,"concrete_test":"Re-run Lemma 4 with q=2 and the Singer set {0,1,3} modulo 7: with F using only two of these three elements, the difference set covers only two nonzero residues, so already s=1 (or any missing residue) has zero representations and Lemma 4 is false as stated. Then verify the corrected lemma with all three Singer elements, replace Corollary 3 by eta_g(n) <= (q_n+1)eta_g(v), and check that the inequalities in Theorem 1 still close, possibly after enlarging N_0. If they close, the paper's first theorem survives with a localized edit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is not Proposition 2 but Lemma 4 in Section 2.3. Lemma 3 (Singer) gives q+1 elements a_0,...,a_q, whose q^2+q ordered differences realize every nonzero residue modulo m = q^2+q+1. The set F in Lemma 4 is defined using only i = 1,...,q. Among the q elements actually used there are only q(q-1) ordered differences, so at least m-1 - q(q-1) = 2q nonzero residues modulo m are not representable as a_i - a_j. Any s whose residue falls in this missing set is not in F - F, so F is not a g-difference basis for [mv]. The proof of Lemma 4 therefore fails as written. Corollary 3 and the Redei-Renyi argument in Theorem 1 depend directly on this construction, making it the load-bearing step for the first main theorem. The natural repair, using all q+1 Singer elements, changes |F| to (q+1)eta_g(v) and requires Corollary 3 to become eta_g(n) <= (q_n+1)eta_g(v); the constants in the proof of Theorem 1 must then be rechecked. This is repairable, but it is not the inequality issue flagged by the reader.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two extremal functions for difference representations in [n]: η_g(n), the minimum size of a set A ⊂ Z with at least g representations of every x ∈ [n], and α_g(n), the maximum size of A ⊂ [n] with at most g representations of every nonzero x. The main results are Theorem 1, asserting that lim_{n→∞} η_g(n)/√n exists and is positive, answering a question of Kravitz, and Theorem 3, asserting that α_g(n) = (1 + o_g(1))√(gn). Theorem 2 gives analogous but parity-dependent limits for η_g(F_p^n)/√(p^n). The proofs use a Rédei–Rényi/Singer construction for the lower bound side of η_g, product and induction constructions for F_p^n, and a Bose–Chowla/Siegel–Walfisz construction for α_g. The paper also surveys sum analogues and applications to coding theory and cryptography.","tokens_in":12550,"tokens_out":8242,"duration_ms":83732,"significance":"If the main theorems hold, they resolve Kravitz's limit-existence question and determine the first-order asymptotic of generalized Sidon sets in [n], improving the Θ(√(gn)) bounds of Xu. The paper connects classical tools — Singer difference sets, Rédei–Rényi's argument, and Bose–Chowla sets — in a clean way, and the applications discussion is useful context. However, two load-bearing constructions contain gaps: Lemma 4 omits one Singer element, and Proposition 2's prime interval does not imply the required N ≤ n. Both gaps appear local and repairable, but the proofs as written do not establish the theorems.","major_comments":[{"comment":"Lemma 4 is false as stated. Singer's Lemma 3 supplies q + 1 elements a_0, ..., a_q whose q^2 + q ordered differences exhaust the nonzero residues modulo m = q^2 + q + 1. The set F is defined with i = 1, ..., q only. The q(q − 1) ordered differences among these q elements cover at most q^2 − q nonzero residues, leaving 2q residues uncovered (namely the differences a_0 − a_i and a_i − a_0). Consequently, in Case 2 of the proof, the assertion that Singer's theorem gives r = a_h − a_ℓ or r − m = a_h − a_ℓ for some h, ℓ is false when the required difference involves the omitted index 0; the proof also silently omits the case r = 0. Since Corollary 3 and the Rédei–Rényi argument of Theorem 1 use exactly the bound |F| = q η_g(v), the proof of Theorem 1 is not valid as written. A repair using all q + 1 Singer elements would change Corollary 3 to η_g(n) ≤ (q_n + 1)η_g(v), and the constants in the proof of Theorem 1 would then need to be rechecked; this appears feasible, but it is not the proof in the manuscript.","section":"Section 2.3, Lemma 4 and Corollary 3"},{"comment":"The displayed prime interval √((1−ε)gn) + 1 ≤ q ≤ √(gn) + 1 does not imply (q^2 − 1)/g ≤ n. For q = √(gn) + 1 one has (q^2 − 1)/g = n + 2√(n/g), which can exceed n. Thus the constructed set of representatives need not lie in [n], and the claimed lower bound α_g(n) ≥ (1 − o(1))√(gn) rests on a stronger prime-selection statement that is not proved. This is also repairable: replacing the upper endpoint with √(gn + 1) gives N ≤ n, and existence of a prime q ≡ 1 (mod g) in the resulting interval follows from Siegel–Walfisz for the fixed modulus g, but the manuscript must state and justify this corrected condition.","section":"Section 4.2, Proposition 2"}],"minor_comments":[{"comment":"Lemma 8 is not self-contained: its statement refers to 'The set A is defined below in the statement of Proposition 2.' It should be reformulated in terms of A_H or in terms of the quotient construction.","section":"Section 4.2, Lemma 8"},{"comment":"In the parenthetical expansion following Equation (10), there is an apparent typo: 'dg+' should read 'd_{2g}'.","section":"Section 4.1, Equation (10)"},{"comment":"The definition of r_{A+A}(s) contains a spurious semicolon: 's = a + a; }' should be 's = a + a}'.","section":"Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's main results are plausible and likely correct after the two repairs indicated above, so I recommend major revision rather than rejection. The errors are concrete and load-bearing, but both appear local: Lemma 4 can be fixed by using all Singer elements at the cost of a constant factor, and Proposition 2 can be fixed by a corrected prime interval. No issues of circularity or fitted parameters were found; the proofs rely on external theorems in a standard way."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's main results are probably correct and worth having, but the proof as written has a concrete hole in Lemma 4 that goes beyond the r=0 omission you flagged. The stress-test note is right: Singer gives q+1 elements, Lemma 4 uses only q of them, so q(q−1) differences cover at most q(q−1) nonzero residues mod m = q²+q+1, leaving 2q residues unrepresented. F is therefore not a g-difference basis for [mv], and the Rédéi–Rényi stepping argument in Theorem 1 loses its foundation. The natural repair — use all q+1 Singer elements — changes |F| to (q+1)η_g(v) and requires Corollary 3 to become η_g(n) ≤ (q_n+1)η_g(v). The extra +1 can be absorbed into the δ slack of the proof, so I'm fairly confident Theorem 1 is salvageable, but the written proof is incorrect as it stands.\n\nWhat is genuinely new: the general-g limit in the integers answers Kravitz's question; the even/odd finite-field limits are new; and the α_g(n) = (1+o_g(1))√(gn) asymptotic sharpens Xu's Θ(√(gn)). The methods are standard — Rédéi–Rényi for the limit, Bose–Chowla for the lower bound — but the combination is a real step within the subfield.\n\nOther soft spots: Proposition 2's prime choice states q ≤ √(gn)+1 and claims N=(q²−1)/g ≤ n. That does not follow; you need q²−1 ≤ gn, i.e., q ≤ √(gn+1). The interval [√((1−ǫ)gn+1), √(gn+1)] still has length proportional to √(gn), so Siegel–Walfisz should deliver the prime; it's a fix, not a dead end. Also, the missing r=0 case in Lemma 4 is real but trivial to handle with the same a_i.\n\nBottom line: the paper is for additive combinatorists who work on difference bases and generalized Sidon sets. It deserves a serious referee, but the referee should insist on a corrected Lemma 4 and a corrected prime interval in Proposition 2. I'd send it out.","headline":"Theorem 1 is likely true but the proof has a real gap in Lemma 4 that the reader's take understates; the paper still deserves a referee.","tokens_in":13167,"tokens_out":5117,"would_cite":false,"duration_ms":46737,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B13","05B10","11B34","11N13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the normalized minimal size of g-difference bases has a positive limit and that maximal g-difference sets have size √(gn).","keywords":["g-difference bases","generalized Sidon sets","difference representation functions","Singer difference sets","Bose-Chowla Sidon sets","additive combinatorics","finite fields","asymptotic extremal bounds"],"falsifier":"Compute, for fixed $g$ and increasingly large $n$, the actual primes $q\\equiv1\\pmod g$ in the interval $[\\sqrt{(1-\\varepsilon)gn}+1,\\sqrt{gn}+1]$; if infinitely many of these intervals contain no prime with $(q^2-1)/g\\le n$, then the embedding step in Proposition 2 fails and the lower-bound proof would need a different construction.","tokens_in":12077,"feed_emoji":"🔢","tokens_out":13671,"duration_ms":136447,"temperature":0.7,"pith_summary":"The paper studies two extremal quantities attached to a set $A$ of integers: the smallest size of $A$ such that every integer $1,\\dots,n$ occurs at least $g$ times as a difference $a_1-a_2$, and the largest size of $A$ such that no nonzero integer occurs more than $g$ times as such a difference. It proves that the first quantity, divided by $\\sqrt{n}$, converges to a positive finite constant, settling an open question from the literature and extending the known $g=1$ result. It proves that the second quantity is $(1+o_g(1))\\sqrt{gn}$, giving the first-order size of generalized Sidon sets. For vector spaces over a fixed finite field it proves that the normalized minima have separate even-dimensional and odd-dimensional limits, and conjectures these limits differ. The upshot is that both the minimal and maximal problems follow the $\\sqrt{n}$ scale, with the multiplicity $g$ entering the maximal problem exactly through a factor $\\sqrt{g}$.","feed_headline":"Limits for g-difference sets pinned at √n and √(gn)","feed_subtitle":"Answers the open limit question and gives the asymptotic √(gn) size for g-repeated differences.","key_machinery":"The argument is carried by two classical constructions plus one weighting identity. For the minimal problem, a perfect difference set in the cyclic group of order $q^2+q+1$ is used to blow up any $g$-difference basis for $[v]$ into one for $[mv]$, while primes in short intervals let the blow-up be placed at the correct scale; iterating this forces $\\liminf$ and $\\limsup$ of $\\eta_g(n)/\\sqrt{n}$ together. For the upper bound on the maximal problem, the central identity is the weighted sum $\\sigma_\\ell = \\sum_{t=1}^{\\ell}\\sum_{i=t+1}^{k}(a_i-a_{i-t})$ of consecutive differences, where cancellation gives an upper bound about $n\\ell(\\ell+1)/2$ and the at-most-$g$ representation condition gives a matching lower bound; choosing $\\ell\\approx\\sqrt{k}$ yields $\\alpha_g(n)\\le(1+o(1))\\sqrt{gn}$. For the lower bound, a Sidon set of size $q$ in $\\mathbb{Z}/(q^2-1)\\mathbb{Z}$ is quotiented by a subgroup of order $g$, producing a set whose nonzero differences each occur at most $g$ times in a cyclic group of size $N=(q^2-1)/g$, which is then embedded into $[n]$.","core_discovery":"On the paper's own terms, the central discovery is that the normalized sequence $\\eta_g(n)/\\sqrt{n}$ has a positive finite limit for every fixed $g$, not merely bounded oscillation, and that the companion maximum $\\alpha_g(n)$ satisfies $\\alpha_g(n)=(1+o_g(1))\\sqrt{gn}$. The first statement answers the existence question posed for generalized difference bases; the second pins down the constant for sets with at most $g$ representations of each nonzero difference. In vector spaces over a finite field, the paper proves that the corresponding even and odd subsequential limits exist and conjectures that they are unequal, signaling a parity effect in that setting.","pith_inferences":["The proof of Proposition 2 asserts that $q\\le\\sqrt{gn}+1$ implies $(q^2-1)/g\\le n$; this implication is not valid in general (for example $g=2,n=3$ gives $q=3$ but $(q^2-1)/g=4>3$), so as written the lower bound relies on a prime selection slightly stronger than the one stated, likely fixable because the required interval still has length growing like $\\sqrt{n}$.","The quotient-of-Sidon-set construction suggests a transfer principle: any group with a difference set of size about $\\sqrt{N}$ should admit $g$-restricted difference sets of size about $\\sqrt{gN}$, which would generalize Theorem 3 to other cyclic and abelian settings.","The even/odd split in $\\mathbb{F}_p^n$ parallels known behaviour for sum-side quantities and hints that the asymptotic constant for vector spaces may depend on whether the dimension is even; testing the conjecture numerically for small $p$ and large $k$ would be a direct check.","A plausible strengthening of Theorem 1 would be an explicit value or bounds for the limiting constant $C_g$; the paper shows existence but leaves its numerical determination open."],"forward_implications":["For every fixed $g$, the minimal $g$-difference basis problem has a true asymptotic scale: $\\eta_g(n)=C_g\\sqrt{n}(1+o(1))$ for some positive constant $C_g$, so comparisons between different $g$ reduce to comparing constants.","The maximal size of a subset of $[n]$ with no difference repeated more than $g$ times is $(1+o(1))\\sqrt{gn}$, settling the first-order behaviour of generalized Sidon sets.","The lower-bound construction gives explicit sets of size $(1-o(1))\\sqrt{gn}$ with the required representation bound, so the asymptotic is achieved rather than merely approached by counting.","In $\\mathbb{F}_p^n$, $\\eta_g$ has separate even and odd limits; if the conjecture is correct, the parity of the dimension is asymptotically visible in the minimal difference-basis size.","The upper-bound method yields a quantitative route: taking $\\ell\\approx\\sqrt{k}$ in the weighted sum controls $\\alpha_g(n)$ to within $o(\\sqrt{gn})$, so the result is not just an order-of-magnitude statement."],"supporting_citations":[{"why":"Poses the question of whether the normalized limit exists; Theorem 1 answers it.","marker":"[20]"},{"why":"Proves the single-representation case that the paper adapts to all g.","marker":"[27]"},{"why":"Outlines the proof of the single-representation case in English; the adaptation follows that outline.","marker":"[22]"},{"why":"Supplies the perfect cyclic difference set used to lift a basis for [v] to one for [mv].","marker":"[29]"},{"why":"Provides primes in short intervals used to choose q_n between the required scales.","marker":"[1]"},{"why":"Constructs the Sidon set in Z/(q^2-1)Z of size q used for the lower bound in Theorem 3.","marker":"[6]"},{"why":"Introduces the quotient-by-subgroup construction and the lemma (A-A) cap H = {0} for the multiplicity bound.","marker":"[19]"},{"why":"Also cited for Lemma 8, the key disjointness property of the quotient construction.","marker":"[15]"},{"why":"Supplies the weighted-sum inequality that yields the upper bound on alpha_g(n).","marker":"[21]"},{"why":"Gives the prior Theta(sqrt(gn)) bounds and the relation that Theorem 3 strengthens to a full asymptotic.","marker":"[31]"}],"fun_headline_variants":["g-difference bases: finite limit exists, constant √g for max sets","Kravitz's question answered: η_g(n)/√n converges","√(gn) asymptotic for at-most-g difference sets","Two limits pinned: η_g and α_g for g-difference sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The lower-bound proof for $\\alpha_g(n)$ assumes that for every large $n$ one can find a prime $q$ with $q\\equiv1\\pmod g$ and $q^2-1\\le gn$, so that the cyclic construction of size $q$ actually sits inside $[n]$; the interval used in the proof only guarantees $q\\le\\sqrt{gn}+1$, which is not by itself enough.","fun_headline_variants_meta":{"raw":{"variants":["g-difference bases: finite limit exists, constant √g for max sets","Kravitz's question answered: η_g(n)/√n converges","√(gn) asymptotic for at-most-g difference sets","Two limits pinned: η_g and α_g for g-difference sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000683,"raw_usage":{"total_tokens":3060,"prompt_tokens":863,"completion_tokens":2197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2120}},"tokens_in":479,"tokens_out":2197,"duration_ms":15320,"temperature":1.0,"reasoning_tokens":2120,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:56:29.767005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for fixed $g$ and increasingly large $n$, the actual primes $q\\equiv1\\pmod g$ in the interval $[\\sqrt{(1-\\varepsilon)gn}+1,\\sqrt{gn}+1]$; if infinitely many of these intervals contain no prime with $(q^2-1)/g\\le n$, then the embedding step in Proposition 2 fails and the lower-bound proof would need a different construction.","supporting_citations":[{"cited_title":"Generalized difference sets and autocorrelation integrals","cited_arxiv_id":"2004.06611","evidence_quote":"Poses the question of whether the normalized limit exists; Theorem 1 answers it."},{"cited_title":"R´ edei and A","cited_arxiv_id":null,"evidence_quote":"Proves the single-representation case that the paper adapts to all g."},{"cited_title":"Mirsky, MathSciNet Review MR003055","cited_arxiv_id":null,"evidence_quote":"Outlines the proof of the single-representation case in English; the adaptation follows that outline."},{"cited_title":"Singer, A theorem in finite projective geometry and so me applications to number theory, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the perfect cyclic difference set used to lift a basis for [v] to one for [mv]."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides primes in short intervals used to choose q_n between the required scales."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the Sidon set in Z/(q^2-1)Z of size q used for the lower bound in Theorem 3."},{"cited_title":"Johnston, M","cited_arxiv_id":null,"evidence_quote":"Introduces the quotient-by-subgroup construction and the lemma (A-A) cap H = {0} for the multiplicity bound."},{"cited_title":"Sidon sets and $C_4$-saturated graphs","cited_arxiv_id":"1810.05262","evidence_quote":"Also cited for Lemma 8, the key disjointness property of the quotient construction."},{"cited_title":"Lindstr¨ om, An inequality for B2-sequences, J","cited_arxiv_id":null,"evidence_quote":"Supplies the weighted-sum inequality that yields the upper bound on alpha_g(n)."},{"cited_title":"Xu, Popular differences and generalized Sidon sets, J","cited_arxiv_id":null,"evidence_quote":"Gives the prior Theta(sqrt(gn)) bounds and the relation that Theorem 3 strengthens to a full asymptotic."}],"review_version":1}