arXiv:2606.28986v1

Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

Yixiu Xiao

math.NT11J2511K3111L07

Paper metadata

Original submission: June 27, 2026
Claimed MSC: 11J25, 11K31, 11L07
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 45 pages

Abstract

Let $k\ge 2$ be fixed. We study the distribution modulo one of the $n^k$ sums $\sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n.$ counted with multiplicity. For $S(h,n) = \sum_{n/2\le a\le n} \mathbf{e}(h\sqrt{a}), \qquad \mathbf{e}(x) = \exp(2πi x),$ we prove second- and fourth-moment estimates matching the diagonal scale up to a factor $n^\varepsilon$ . More precisely, $\sum_{H/2\le h\le H} \left| S(h,n) \right|^2 \ll_{\varepsilon,δ} Hn^{1+\varepsilon}$ uniformly for $H\ge n^{1/2+δ}$ , and $\sum_{H/2\le h\le H} \left| S(h,n) \right|^4 \ll_{\varepsilon,δ} Hn^{2+\varepsilon}$ uniformly for $n^{1/2+δ} \le H \le n^{2/3}$ , where $0<δ<1/6$ in the fourth-moment estimate. Combining the second-moment bound with pointwise exponential-sum estimates and the Erdős--Turán inequality, we obtain $D_k(n) \le n^{-ρ_k+o(1)}, \qquad ρ_k = \frac{71k+26}{26k+116},$ as $n\to\infty$ , where $D_k(n)$ denotes the discrepancy with respect to arbitrary subintervals of $[0,1)$ .

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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 15, 2026

01Statements3 reported findingsCorrect

The second- and fourth-moment estimates for dyadic sums of square roots and the resulting discrepancy exponent are correct as stated.

Theorem 1.1Correct

Second-moment estimate

Pages 4 and 33–36 · Theorem 1.1 and its proof · arXiv:2606.28986v1

For $H\geq n^{1/2+\delta}$ , the diagonal contribution has size $Hn$ , while smoothing and Poisson summation convert the off-diagonal contribution into a weighted quadratic dual sum. Proposition 3.11 bounds that dual second moment by the same scale up to $n^\varepsilon$ . The stationary and nonstationary ranges cover every frequency introduced by the partition, and their error terms are absorbed uniformly in the printed range. This gives $M_2(H,n)\ll_{\varepsilon,\delta}Hn^{1+\varepsilon}$ .

Full paper, version 1 ↗

Theorem 1.2Correct

Fourth-moment estimate

Pages 4 and 36–40 · Theorem 1.2 and its proof · arXiv:2606.28986v1

In the range $n^{1/2+\delta}\leq H\leq n^{2/3}$ , the fourth power is dualized after the same dyadic smoothing. Proposition 3.14 controls the resulting additive energy through the reciprocal-spacing estimate and the divisor and greatest-common-divisor sums of Lemmas 3.12–3.13. Substitution of the admissible parameter ranges leaves $Hn^{2+\varepsilon}$ as the dominant term, proving the stated bound.

Full paper, version 1 ↗

Theorem 1.3Correct

Discrepancy exponent

Pages 4 and 41–44 · Theorem 1.3 and its proof · arXiv:2606.28986v1

The Erdős--Turán inequality reduces discrepancy to dyadic ranges of the exponential sums $S(h,n)^k$ . The second-moment estimate handles the range beginning at $n^{1/2+\delta}$ , and the cited pointwise estimate handles the complementary frequencies. Balancing the truncation and exponential-sum terms gives $\rho_k=\frac{71k+26}{26k+116},$ with the limiting endpoint obtained by letting the auxiliary losses tend to zero. The resulting estimate is $D_k(n)\leq n^{-\rho_k+o(1)}$ for each fixed $k\geq2$ .

Full paper, version 1 ↗

02Proofs4 reported findingsCorrect

The near-collision bounds, smoothing and Poisson analysis, weighted dual moment estimates, and final Erdős--Turán optimization are correct and complete in the stated parameter ranges.

Lemmas 2.6–2.8Correct and complete

Root counts and square-root near collisions

Pages 7–12 · Lemmas 2.6–2.8 · arXiv:2606.28986v1

The congruence count is multiplicative and has the stated prime-power bounds. After separating the diagonal, the spacing of square roots turns each near-collision condition into an interval count of the claimed length. Summation over the divisor parameters then gives the elementary second-moment estimate with the printed $n^\varepsilon$ loss. Endpoint and repeated-root cases are included in these counts.

Lemmas 3.1–3.10Correct and complete

Smoothing, Poisson summation, and phase analysis

Pages 15–33 · Section 3 through Proposition 3.6 · arXiv:2606.28986v1

The smooth dyadic partitions preserve the original sums with bounded overlap. Poisson summation produces the stated dual phases, and the stationary point lies in the support exactly in the retained frequency range. Repeated integration by parts controls the remaining frequencies. The derivative sizes, Hessian factors, and weights agree with the powers of $H$ and $n$ used in both moment arguments.

Propositions 3.11 and 3.14Correct and complete

Dual second- and fourth-moment estimates

Pages 33–40 · Propositions 3.11 and 3.14 · arXiv:2606.28986v1

For the quadratic dual phase, differencing reduces the second moment to a root count modulo the relevant modulus, with the diagonal treated separately. The fourth moment is reduced to reciprocal additive energy; Lemmas 3.12–3.13 supply the needed energy and greatest-common-divisor bounds. Inserting the weights and summing the dyadic parameters gives precisely the two moment estimates without an uncovered parameter regime.

Proof of Theorem 1.3Correct and complete

Discrepancy optimization

Pages 41–44 · Section 4 · arXiv:2606.28986v1

The dyadic decomposition of the Erdős--Turán sum uses the moment estimate only where its lower threshold holds and uses the pointwise bound elsewhere. The selected truncation parameter lies in every required range. Equating the resulting exponents yields $(71k+26)/(26k+116)$ , and all logarithmic factors are legitimately absorbed into $n^{o(1)}$ .

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.28986v1
Authors listed: Yixiu Xiao
Audit date: August 15, 2026

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Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Second-moment estimate

Fourth-moment estimate

Discrepancy exponent

Root counts and square-root near collisions

Smoothing, Poisson summation, and phase analysis

Dual second- and fourth-moment estimates

Discrepancy optimization

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