{"id":"f11681b5-b9a9-4039-922b-9669b7d3e94f","arxiv_id":"2411.17823","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For modular inverses with moduli up to X, the paper proves quantitative box, ball, and convex discrepancy bounds, powered by a new triple Kloosterman sum estimate.","lead":"This paper gives new upper and lower bounds for how evenly the points formed by modular inverses fill the unit square when the modulus is allowed to grow. The proof introduces a sharper average estimate for triple sums of Kloosterman sums, which may be useful beyond this problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The smoothing weight in Section 4.4 is inverted: with V_{n,\\Delta}(x)=W_\\Delta(xX/(4\\pi\\sqrt n)) the smooth sum localizes to c\\in[X/2,X], not c\\in[X,2X], so the displayed \\Delta^2X comparison for reducing to the smooth second moment is invalid as written.","rationale":"I read the paper as a coherent forward chain: a new second moment bound via Kuznetsov and the spectral large sieve, a triple-sum bound via Selberg identity, and discrepancy estimates from exponential sum bounds. The reader's flagged assumption, the imported hybrid spectral large sieve of Lemma 4.3, is a cited theorem and I do not see an independent correctness risk there. However, the written proof of Proposition 4.1 contains an internal inconsistency in the smoothing step: the weight V_{n,\\Delta} as defined does not approximate the interval [X,2X] appearing in K^{(2)}. This is a load-bearing issue because Proposition 4.1 is the basis for Theorems 1.3 and 1.1. The defect is almost certainly a typographical inversion of the argument of W_\\Delta, and the subsequent Mellin-inversion and dyadic estimates are compatible with the corrected weight. I therefore would not reject the mathematics, but the manuscript as currently written needs a correction before the claim can be taken as proved.","tokens_in":20885,"tokens_out":22545,"duration_ms":186712,"concrete_test":"Evaluate the defining identity at the endpoints: with x=4\\pi\\sqrt n/c, the current definition gives V_{n,\\Delta}(4\\pi\\sqrt n/X)=W_\\Delta(1)=1 but V_{n,\\Delta}(4\\pi\\sqrt n/(2X))=W_\\Delta(1/2)=0, even though 1_{X\\le c<2X}=1 at both points. Recompute the first inequality in Section 4.4 with the corrected definition V_{n,\\Delta}(x)=W_\\Delta(4\\pi\\sqrt n/(xX)); if Proposition 4.1 then follows unchanged, the issue is a typographical inversion and acceptance should be conditional on the correction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 4.1 is the engine for Theorem 1.1, and its proof in Section 4.4 reduces the sharp cut-off sum to the smooth sum K_*^{(2)}. The reduction uses the inequality |\\sum_{X\\le c<2X} c^{-1}S(n,\\pm1,c)|^2 \\le 2|\\sum_c c^{-1}S(n,\\pm1,c)(1_{X\\le c<2X}-V_{n,\\Delta}(4\\pi\\sqrt n/c))|^2 + 2K_*^{(2)} \\le \\Delta^2X + K_*^{(2)}. But V_{n,\\Delta}(4\\pi\\sqrt n/c)=W_\\Delta(X/c). Since W_\\Delta=1 on [1,2] and has support [1-\\Delta,2+\\Delta], this smooth function is 1 roughly for c\\in[X/2,X] and is 0 for c\\in[X,2X] except near c=X. Thus 1_{X\\le c<2X}-V_{n,\\Delta}(4\\pi\\sqrt n/c) is not O(\\Delta) on [X,2X]; the unweighted contribution over most of the interval is not captured, and the claimed \\le\\Delta^2X bound fails. The fix is almost certainly to define V_{n,\\Delta}(x)=W_\\Delta(4\\pi\\sqrt n/(xX)), or to use the interval [X/2,X], but as written the proof of the paper's central estimate is internally inconsistent.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the distribution of the modular hyperbola points (a/c, b/c) with ab ≡ 1 mod c and c ≤ X, viewed as a set in the torus. The main analytic engine is a new bound for the triple Kloosterman sum K(M,N;X), proved through a second-moment estimate for sums of Kloosterman sums. This estimate is obtained from the Kuznetsov formula, a hybrid spectral large sieve, and a dyadic/mellin parameter optimization. From the triple sum bound the authors derive quantitative discrepancy bounds for boxes, balls, and convex sets, together with lower bounds that exhibit deviations from random point sets. The paper is clearly written and the overall strategy is coherent, but the proof of the central second-moment estimate contains a localized error in the smoothing construction that must be corrected.","tokens_in":21169,"tokens_out":23286,"duration_ms":187786,"significance":"If the main results are correct, they constitute a substantial advance: the bound K(M,N;X) ≤ (MNX + (MN)^{2/3}X^{7/6})(MNX)^{o(1)} is stronger than what follows from individual Sarnak–Tsimerman/Kıral bounds, even under the Ramanujan–Petersson conjecture, and it yields power-saving discrepancy estimates for the union of modular hyperbolas, quantitatively improving and complementing the qualitative equidistribution results of Selberg and Good. The lower bounds showing deviations from random sets are also of independent interest. The derivation is forward and does not rely on fitted parameters; the paper makes honest use of standard external machinery (Kuznetsov formula, spectral large sieve, Selberg identity, Barton–Montgomery–Vaaler, Harman, Schmidt) and gives explicit parameter balances. The main weakness is a local but load-bearing error in Section 4.4 that currently invalidates the proof of Proposition 4.1.","major_comments":[{"comment":"The smoothing weight is inverted. With V_{n,Δ}(x)=W_Δ(xX/(4π√n)) one has V_{n,Δ}(4π√n/c)=W_Δ(X/c). Since W_Δ is identically 1 on [1,2] and supported on [1−Δ,2+Δ], this weight is approximately 1 for c∈[X/2,X] and vanishes for c>X/(1−Δ), i.e. for almost all of [X,2X]. Hence the function 1_{[X,2X]}(c)−V_{n,Δ}(4π√n/c) is not O(Δ) on [X,2X]; the unweighted contribution over most of [X,2X] is of size comparable to X, not Δ²X. The displayed inequality that precedes (4.7), asserting that this contribution is bounded by Δ²X, is therefore false, and the reduction of the sharp-cutoff second moment to the smooth quantity K_*^{(2)} does not prove Proposition 4.1 as written. The natural fix is to define V_{n,Δ}(x)=W_Δ(4π√n/(xX)), so that V_{n,Δ}(4π√n/c)=W_Δ(c/X), or equivalently to work with the interval [X/2,X]; the subsequent Mellin and spectral argument appears to be compatible with either correction. Since Proposition 4.1 is the engine for Theorems 1.3 and 1.1, this point must be repaired before the paper can be accepted.","section":"Section 4.4, the paragraph before Eq. (4.7)"}],"minor_comments":[{"comment":"The phrase “As the referee remarked” is inappropriate in a preprint or submitted version; it should be removed or rephrased as a neutral acknowledgement of the alternative approach.","section":"Remark 2.5"},{"comment":"The final integration is stated as giving N+Δ⁻¹, but the N-term produces a factor log(1/Δ) that is then absorbed by the (NX)^{o(1)} notation; for precision this should be acknowledged in the displayed estimate.","section":"Section 4.4, after Eq. (4.11)"},{"comment":"In the derivation of Theorem 1.1 from Theorem 1.3, the passage from the double sum over m,n to a single sum over r uses the divisor bound implicitly; the authors should state explicitly that the number of representations r=m'n' with m' and n' in dyadic intervals is at most (MN)^{o(1)}.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The only mathematical obstacle I found is the inverted smoothing weight in Section 4.4; it is a genuine error in the proof of the paper's central estimate, but it is clearly localized and has an obvious fix. If the authors redefine the weight as W_Δ(4π√n/(xX)) and re-check the Mellin estimates, I expect the rest of the proof to stand. I therefore recommend major revision rather than rejection. I also note the unusual self-referential phrase in Remark 2.5, which should be cleaned up."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. The paper is worth serious attention, but the proof of the central estimate has a concrete error that needs fixing. Theorem 1.1 is a genuine improvement: it removes the Ramanujan–Petersson exponent from the average triple-sum bound and replaces (MN)^{1+θ}X^{7/6} with (MN)^{2/3}X^{7/6}. The discrepancy exponents for boxes and convex sets are new, and the log X / X lower bound for convex discrepancy is a neat observation. The chain from second moment to triple sum to discrepancy is well organized, and the heavy analytic machinery (Kuznetsov, spectral large sieve, Selberg identity, Barton–Montgomery–Vaaler, Harman) is used in a way that looks legitimate.\n\nThe soft spot is Section 4.4. As written, V_{n,Δ}(4π√n/c) = W_Δ(X/c), so the smooth sum is supported near c ∈ [X/2, X], not c ∈ [X, 2X]. The claimed inequality |Σ_{X≤c<2X} c^{-1}S(n,±1,c)|^2 ≤ Δ^2X + ... is therefore not true: on most of [X, 2X] the difference between the sharp cutoff and the smooth weight is of size 1, not O(Δ). This is a load-bearing step in Proposition 4.1. I think it is almost certainly a typo — the natural fix is to define V_{n,Δ}(x) = W_Δ(4π√n/(xX)), or to run the dyadic argument on [X/2, X] — but as submitted the central estimate does not follow.\n\nThe reader's soundness score of 8 is too generous; I would call the paper promising but in need of a correction. The other flagged issue, the deferred polygon approximation in Lemma 3.1, is minor and standard. The reliance on the hybrid spectral large sieve is heavy but properly cited. The paper is not circular and has no fitted parameters.\n\nVerdict: send to a serious referee, but tell the authors to fix the smoothing weight and re-check the dyadic reduction before final acceptance. The paper is aimed at analytic number theorists working on Kloosterman sums and uniform distribution; those readers will get clear value once the correction is made. For a reading group I would want to see the corrected version first.","headline":"Central proof has a fixable smoothing-weight inversion; otherwise this is a strong, important paper with genuine new exponents.","tokens_in":659,"tokens_out":950,"would_cite":true,"duration_ms":57928,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K38","11L05","11F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a bound for triple sums of Kloosterman sums and derives from it that modular-inverse points modulo c up to X are equidistributed with explicit power-saving discrepancy bounds, while lower bounds exhibit non-randomness.","keywords":["modular inverses","Kloosterman sums","discrepancy","equidistribution","Kuznetsov formula","spectral large sieve","automorphic forms","modular hyperbola"],"falsifier":"Compute the normalized second moment $\\sum_{\\pm}\\sum_{N\\le n<2N}|\\sum_{X\\le c<2X}S(n,\\pm1;c)/c|^2$ for $N=X$ and increasing $X$; Proposition 4.1 predicts it is $\\ll N^{4/3+o(1)}$, so any sequence exceeding this would refute the central chain. On the distribution side, one can search discs of radius $R=X^{-1/2+\\varepsilon}$ around rational points with small denominators and check whether the counts stay within $(X^{-1}+R^{2/3}X^{-2/3})X^{o(1)}$ of $\\pi R^2 N(X)$.","tokens_in":20645,"feed_emoji":"📐","tokens_out":18088,"duration_ms":144372,"temperature":0.7,"pith_summary":"The paper studies the set $S(X)$ of pairs $(a/c,b/c)$ in the unit square with $ab\\equiv 1 \\bmod c$ and $c\\le X$, and asks how evenly these modular-inverse points are distributed as $X$ grows. Its central claim is that the discrepancy of $S(X)$ — the largest relative error between the fraction of points in a box, disc, or convex set and its area — decays by an explicit power of $X$: at most $X^{-5/6+o(1)}$ for boxes, $X^{-2/3+o(1)}$ for discs, and $X^{-11/24+o(1)}$ for convex sets. It also proves lower bounds of order $1/X$ for boxes and discs and $(\\log X)/X$ for convex sets, so the set is not perfectly random at scales of size $1/X$. The new ingredient is an unconditional bound for a triple sum of Kloosterman sums that averages over $m$ and $n$ and is stronger than what individual pointwise bounds would give. The paper thus supplies quantitative forms of the classical equidistribution of modular inverses and traces small-scale deviations from randomness to arithmetic structure near rational points.","feed_headline":"Modular-inverse points miss boxes by no more than X^{-5/6}","feed_subtitle":"An average Kloosterman bound yields box, disc, and convex-set discrepancy exponents for modular inverses.","key_machinery":"The central object is the second moment $K^{(2)}(N;X)=\\sum_{N\\le |n|<2N}|\\sum_{c\\le X}S(n,1;c)|^2$ and the derived triple sum $K(M,N;X)$. The mechanism is spectral: the Kuznetsov formula rewrites sums of Kloosterman sums $S(m,n;c)$ as spectral averages of Fourier coefficients $\\rho_\\varpi(m)\\rho_\\varpi(n)$ with Bessel-transform weights; Lemma 4.2 bounds those integral transforms, and the hybrid spectral large sieve (an averaging inequality for Fourier coefficients over frequency windows) controls the resulting spectral average by $(TH+N)^{1+\\varepsilon}$. Then Selberg's identity reduces the triple sum to dyadic second moments, and on the distribution side, discrepancy bounds follow from exponential-sum criteria for boxes, balls, and well-shaped convex sets.","core_discovery":"On its own terms, the discovery is the inequality $K(M,N;X)\\le (MNX+(MN)^{2/3}X^{7/6})(MNX)^{o(1)}$ for the triple sum of Kloosterman sums, obtained from the second-moment bound $K^{(2)}(N;X)\\le (NX^2+N^{1/3}X^{7/3})(NX)^{o(1)}$. The proof rewrites sums of Kloosterman sums through the Kuznetsov formula, estimates the relevant Bessel integral transforms, and applies the hybrid spectral large sieve; an identity of Selberg then passes from second moments to the full triple sum. The distribution consequences are the discrepancy bounds $\\Delta(X,B)\\le X^{-5/6+o(1)}$, $\\Delta(X,D)\\le X^{-2/3+o(1)}$, and $\\Delta(X,C)\\le X^{-11/24+o(1)}$, together with lower bounds $\\Delta(X,B),\\Delta(X,D)\\gg 1/X$ and $\\Delta(X,C)\\gg (\\log X)/X$. These lower bounds exhibit deviations from random point sets at small scales, and the upper bounds depend on no unproved exponent toward the Ramanujan–Petersson conjecture.","pith_inferences":["A natural extension not pursued in the paper is to interpolate between the sharp and smooth cutoffs by letting the transition width of the weight $W_\\Delta$ decay slowly; the proof indicates the $X^{7/6}$ term in Theorem 1.1 is a cutoff artifact, so one would obtain a continuously varying box-discrepancy exponent between $-5/6$ and $-1$.","Because the triple-sum bound is unconditional and averaged, the same spectral strategy may apply to other families of arithmetic point sets, such as solutions of $a y\\equiv b\\bmod c$ with varying congruences, where individual exponential-sum bounds are conditional but averaged bounds might still be provable.","One could test the predicted small-scale deviation by sampling $S(X)$ for moderate $X$ and measuring discrepancies of boxes centered at rational points with small denominators; the lower-bound construction suggests the cellular structure near such points, not random fluctuation, drives the $1/X$ and $(\\log X)/X$ errors."],"forward_implications":["Any box of volume $\\mu(B)>X^{-1}$ contains $\\mu(B)N(X)(1+O(\\mu(B)^{-1/3}X^{-1/3+o(1)}))$ points of $S(X)$, so small boxes are controlled more sharply than the global box discrepancy.","Discs of radius $R\\ge X^{-1/2+o(1)}$ satisfy $\\#(D\\cap S(X))/N(X)=\\pi R^2+O((X^{-1}+R^{2/3}X^{-2/3})X^{o(1)})$, giving ball discrepancy $\\Delta(X,D)\\le X^{-2/3+o(1)}$.","Convex sets have isotropic discrepancy at most $X^{-11/24+o(1)}$, while the lower bound $(\\log X)/X$ shows the set is not random at scales of order $1/X$.","Replacing the sharp cutoff $c\\le X$ by a smooth weight improves the box discrepancy to $X^{-1+o(1)}$, the order that the modified Selberg–Linnik conjecture would supply.","The triple-sum bound is uniform in $M,N$ and does not depend on any unproved spectral exponent, so the discrepancy exponents are unconditional."],"supporting_citations":[{"why":"States the Kuznetsov formula in the precise form used to turn sums of Kloosterman sums into spectral averages over Fourier coefficients.","marker":"[20]"},{"why":"Supplies the spectral large sieve for equal window and scale, the backbone of Lemma 4.3.","marker":"[9]"},{"why":"Provides the hybrid spectral large sieve with general window size H that Proposition 4.1 needs, including the continuous spectrum.","marker":"[21]"},{"why":"Extends the hybrid large sieve to holomorphic forms, completing the spectral coverage required in Lemma 4.3.","marker":"[28]"},{"why":"Provides the box-counting upper and lower estimates used to prove the small-box discrepancy bound.","marker":"[3]"},{"why":"Supplies the ball-discrepancy estimate that converts exponential-sum bounds into disc counting and ball discrepancy.","marker":"[18]"},{"why":"Gives the well-shaped set framework and dyadic square approximation used for the isotropic discrepancy bound.","marker":"[35]"}],"fun_headline_variants":["Triple Kloosterman bound sharpens modular inverse discrepancy","Modular inverses: new discrepancy bounds from Kloosterman sums","Kloosterman triple sum yields sharper discrepancy exponents","Modular inverse deviations traced to Kloosterman triple sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the hybrid spectral large sieve inequality is valid in exactly the imported form, with the $(TH+N)^{1+\\varepsilon}$ factor and covering the continuous spectrum and holomorphic forms, because the second-moment bound, the triple-sum bound, and every discrepancy exponent derived from them rest on it.","fun_headline_variants_meta":{"raw":{"variants":["Triple Kloosterman bound sharpens modular inverse discrepancy","Modular inverses: new discrepancy bounds from Kloosterman sums","Kloosterman triple sum yields sharper discrepancy exponents","Modular inverse deviations traced to Kloosterman triple sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1249,"prompt_tokens":845,"completion_tokens":404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":336}},"tokens_in":461,"tokens_out":404,"duration_ms":3975,"temperature":1.0,"reasoning_tokens":336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:48:29.292504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the normalized second moment $\\sum_{\\pm}\\sum_{N\\le n<2N}|\\sum_{X\\le c<2X}S(n,\\pm1;c)/c|^2$ for $N=X$ and increasing $X$; Proposition 4.1 predicts it is $\\ll N^{4/3+o(1)}$, so any sequence exceeding this would refute the central chain. On the distribution side, one can search discs of radius $R=X^{-1/2+\\varepsilon}$ around rational points with small denominators and check whether the counts stay within $(X^{-1}+R^{2/3}X^{-2/3})X^{o(1)}$ of $\\pi R^2 N(X)$.","supporting_citations":[{"cited_title":"Iwaniec and E","cited_arxiv_id":null,"evidence_quote":"States the Kuznetsov formula in the precise form used to turn sums of Kloosterman sums into spectral averages over Fourier coefficients."},{"cited_title":"Deshouillers and H","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral large sieve for equal window and scale, the backbone of Lemma 4.3."},{"cited_title":"Jutila, ‘On the spectral large sieve inequalities’, Funct","cited_arxiv_id":null,"evidence_quote":"Provides the hybrid spectral large sieve with general window size H that Proposition 4.1 needs, including the continuous spectrum."},{"cited_title":"Lam, ‘A local large sieve inequality for cusp forms’, J","cited_arxiv_id":null,"evidence_quote":"Extends the hybrid large sieve to holomorphic forms, completing the spectral coverage required in Lemma 4.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the box-counting upper and lower estimates used to prove the small-box discrepancy bound."},{"cited_title":"Harman, ‘On the Erd˝ os–Tur´ an inequality for balls’,Acta Arith","cited_arxiv_id":null,"evidence_quote":"Supplies the ball-discrepancy estimate that converts exponential-sum bounds into disc counting and ball discrepancy."},{"cited_title":"Schmidt, ‘Irregularities of distribution","cited_arxiv_id":null,"evidence_quote":"Gives the well-shaped set framework and dyadic square approximation used for the isotropic discrepancy bound."}],"review_version":1}