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Triple sums of Kloosterman sums and the discrepancy of modular inverses

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Pith's one-line read 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.

desk verdict Central proof has a fixable smoothing-weight inversion; otherwise this is a strong, important paper with genuine new exponents. read the letter →

arxiv 2411.17823 v3 pith:MDKII4D7 submitted 2024-11-26 math.NT

Valentin Blomer , Morten S. Risager , Igor E. Shparlinski This is my paper · ORCID
classification math.NT MSC 11K3811L0511F12
keywords modularinversesKloostermansumsdiscrepancyequidistributionKuznetsovformulaspectrallargesieveautomorphicformshyperbola
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\pi R^2 N(X)$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

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.

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 (1)
  1. [Section 4.4, the paragraph before Eq. (4.7)] 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.
minor comments (3)
  1. [Remark 2.5] 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.
  2. [Section 4.4, after Eq. (4.11)] 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.
  3. [Section 4.1] 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)}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central estimate rests on external spectral large sieve and Kuznetsov machinery; there is no fitted input or self-citation chain.

full rationale

The claimed derivation chain is forward and externally anchored. Theorem 1.1 is deduced in Section 4.1 from Theorem 1.3 via Selberg's identity and divisor-function bounds; Theorem 1.3 is obtained from Proposition 4.1 by partial summation. Proposition 4.1 is proved in Section 4.4 using the Kuznetsov formula stated in Lemma 4.3, with the spectral large sieve imported from Deshouillers-Iwaniec, Jutila [21, Theorem 1.1], and Lam [28], which are independent external results; the dual large sieve (Lemma 4.4) and integral estimates (Lemma 4.2) are proved in the paper. No parameter is fitted to data and no 'prediction' is re-introduced as an input. The only self-citations, e.g. [23] and [40] in the isotropic discrepancy proof, are to prior techniques for approximating well-shaped sets and are not used to replace an argument: the proof of Theorem 1.9 is carried out with the packing estimates (3.5)-(3.6), Schmidt-style well-shapedness, and Theorems 1.6-1.7. The possible inversion of the smoothing cutoff in Section 4.4 noted by the referee-style check is a correctness concern, not a circularity: even if the displayed Delta^2 X reduction fails, that would make the proof erroneous rather than circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results rest entirely on standard theorems from the analytic number theory literature; the paper introduces no new objects and fits no empirical constants. Auxiliary proof parameters such as Δ=(NX)^{-1/3}, L, M and N are chosen by asymptotic balancing and do not appear as assumptions in the final statements.

assumptions (6)
  • standard math Kuznetsov formula for SL(2,Z) together with the spectral large sieve inequality (Lemma 4.3), including the continuous spectrum and holomorphic forms with the (TH+N)^{1+ε} bound.
    Imported from [20, Theorems 16.5-16.6], [9], [21, Theorem 1.1] and [28]; used in Section 4.3 to convert the second moment of Kloosterman sums into a spectral expression.
  • standard math Weil bound |S(m,n;c)| ≤ gcd(m,n,c)^{1/2} c^{1/2} τ(c).
    Used in Section 4.4 to control the error introduced by replacing the sharp cutoff c≤X with a smooth function; cited to [20, Corollary 11.12].
  • standard math Selberg identity for Kloosterman sums, decomposing S(n,±m,c) via d|gcd(m,n,c) into S(mn/d^2,±1,c/d).
    Used in Section 4.1 to reduce the triple sum over m,n,c to second-moment data; cited to [1,17,27,36,45].
  • standard math Asymptotic N(X)=∑_{c≤X} φ(c) = (3/π^2+o(1))X^2.
    Used throughout as the normalization for discrepancy; cited to [44, Satz 1] and [30].
  • standard math Koksma-Szüsz inequality and Barton-Montgomery-Vaaler box estimates, as stated in Lemmas 2.1 and 2.2.
    Imported from [25,42] and [3]; used in Sections 2.2, 5.1 and 5.2 to turn exponential sum bounds into discrepancy bounds for boxes.
  • standard math Harman's ball discrepancy lemma (Lemma 2.4) and Schmidt's well-shaped approximation machinery including the convex-set well-shaped constant 4+π.
    Harman's lemma is from [18, Theorem 2]; the well-shaped framework is from [35, Section 3] and uses perimeter facts from [41]; used for disc and isotropic discrepancy bounds.

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Pith. "Pith review of Triple sums of Kloosterman sums and the discrepancy of modular inverses." pith.science (2026). https://pith.science/paper/MDKII4D7

@misc{pith2026241117823,
  author       = {Pith},
  title        = {Pith review of: Triple sums of Kloosterman sums and the discrepancy of modular inverses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDKII4D7}},
  note         = {Machine review of arXiv:2411.17823}
}
abstract

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums.

Figures

Figures reproduced from arXiv: 2411.17823 by the authors.

Figure 1.1
Figure 1.1. Modular inverses: X “ 600, NpXq “ 109500. ‚ the box r0, 1s ˆ r0, 1{p2Xqs contains no point from SpXq; ‚ the box r2{ ? X, 3{ ? Xs ˆ r0.2{ ? X, 0.25{ ? Xs contains no point from SpXq; ‚ the box r0, 1{ ? Xs ˆ r0, 1{ ? Xs contains exactly the p1 ` op1qqX points (1.7) while by (1.3) the expected value is only p3{π 2 ` op1qqX . Similar phenomena occur at other rational points with small denominators. From this discussion … view at source ↗
Figure 3.1
Figure 3.1. Ω and Bpiq in U for i ď 6 Using again (3.4) and (3.2) we see also that if Ω is η-well-shaped then (3.6) 0 ď µpΩq ´ ÿn i“1 ÿ ΓPBi µpΓq ď η2 ´n`1{2 . 3.2. Approximations of convex sets. We now assume that Ω Ď U is convex. In this case the closure Ω, interior Ω˝ , restrictions and extensions Ω˘ε , Ωr´ε are all convex. Also recall that the boundary of a convex set has measure zero; see [29, Theorem 1]. Lemma 3.1. A conv… view at source ↗
Figure 3.2
Figure 3.2. Convex polygon P, P with inner rectangles, P with outer rectangles and circular sectors. For the inside rectangles, the convexity of P implies that these overlap and their union covers Ω´ ε [PITH_FULL_IMAGE:figures/full_fig_p012_3_2.png] view at source ↗

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