Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

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

• The paper derives optimal moment bounds and an explicit decay rate for the discrepancy of sums of square roots modulo one.
• It introduces smoothing and duality frameworks that effectively control exponential sum cancellations and align with the expected random model.
• The results yield the sharpest known equidistribution rates for k ≥ 2, with significant implications for numerical integration and Diophantine approximation.

Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

Introduction and Background

The paper investigates the distribution modulo one of sums of square roots of integers, specifically for sums of the form $\sqrt{a_1}+\cdots+\sqrt{a_k}$ where $1 \leq a_i \leq n$ and $k\geq 2$ is fixed. The core object of study is the global distribution of the fractional parts of these sums in the unit interval, quantified via the discrepancy $D_k(n)$ with respect to arbitrary subintervals of $[0,1)$. In contrast with earlier works, which explored inhomogeneous Diophantine approximation (e.g., minimal distance to a prescribed target), this paper tackles the uniformity of distribution modulo one, highlighting the analytic challenge posed by requiring uniform control of exponential sums over many frequencies.

Main Results

The central contributions are high-precision moment estimates for exponential sums of the type $S(h, n) = \sum_{n/2 \leq a \leq n} e(h\sqrt{a})$, leading to explicit discrepancy bounds for the sums of $k$ square roots. The main theorems are:

1. Second Moment Bound: For any $\delta>0$ and $\varepsilon>0$, uniformly for $H \geq n^{1/2+\delta}$,

$1 \leq a_i \leq n$0

This matches the expected "diagonal scale" up to a sub-polynomial factor.

2. Fourth Moment Bound: For $1 \leq a_i \leq n$1 and $1 \leq a_i \leq n$2, uniformly for $1 \leq a_i \leq n$3,

$1 \leq a_i \leq n$4

This provides the first unconditional step towards higher even moments, leveraging smoothing techniques and analysis on the dual side via Poisson summation.

3. Discrepancy Estimate: The two moment bounds, together with careful application of the Erdős–Turán inequality and sharp pointwise exponential sum estimates, yield that for fixed $1 \leq a_i \leq n$5,

$1 \leq a_i \leq n$6

This provides a power-saving, uniform upper bound for the distribution discrepancy in subintervals, where previous methods failed to deliver analogous results for the global distribution problem.

Analytic Approach and Techniques

The analysis proceeds through several key steps:

• Smoothing and Duality: The initial sharp exponential sums are replaced with smoothed analogues to localize analysis away from boundary artifacts. Poisson summation transforms the sum over $1 \leq a_i \leq n$7 into a dual quadratic sum over $1 \leq a_i \leq n$8, producing manageable main terms and sharply bounded error terms.
• Stationary Phase and Non-stationary Regimes: Asymptotic evaluation leverages stationary phase analysis for the main range of $1 \leq a_i \leq n$9 contributing to the dual sum, and integration by parts for the outlying non-stationary region, yielding precise uniform bounds.
• Moment Reductions: Exploiting the dual sum expansion, $k\geq 2$0-th moments are reduced to combinatorial "energy" counts (quasi-diagonal and off-diagonal) among tuples of the form $k\geq 2$1. The key innovation is matching or approaching the conjectural random behavior, up to a factor $k\geq 2$2.
• Exponent Pairs and Weyl Sums: Optimal use of modern exponent pairs for exponential sums (notably, Bourgain's $k\geq 2$3 pair) allows for precise balancing between pointwise and mean estimates, crucial in the regime of large $k\geq 2$4 and $k\geq 2$5.
• Discrepancy via Erdős–Turán: The mean-square bounds on exponential sums, combined with Erdős–Turán, propagate to discrepancy bounds, with a detailed dyadic argument capturing the full parameter space.

Strong Numerical Claims and Contrasts

A particularly strong quantitative claim is the explicit exponent $k\geq 2$6 in the discrepancy estimate, which is unattainable by previous methods focused on inhomogeneous approximation and reflects careful balancing of moment and pointwise bounds. The estimates are shown to be optimal (modulo $k\geq 2$7) in certain parameter ranges, with elaboration on the limitations and the expected shape of higher moments.

Additionally, the paper introduces a conjecture for higher moment bounds (Conjecture: Weighted dual quadratic large sieve), predicting that $k\geq 2$8-th moments should behave essentially like those of $k\geq 2$9 independent random points, up to $D_k(n)$0.

Theoretical and Practical Implications

On the theoretical side, the results advance the quantitative understanding of the uniform distribution properties of additive functions involving roots, a theme central in both analytic number theory and applications in computational complexity. The explicit moment and discrepancy bounds open avenues for:

• Improved quantitative metric results in the distribution of algebraic numbers modulo one.
• Tighter lower bounds for separation problems in fields such as algorithmic number theory and symbolic computation.

On the practical side, quantifying how sums involving square roots cluster or spread modulo one has direct implications for separation of floating-point computations, pseudorandomness in algorithmic settings, and lower bound constructions in computational geometry.

Future Directions

The paper lays groundwork for several future developments:

• Extending to Higher Moments: The conjectural estimates for higher even moments suggest further analysis, either via combinatorial reductions or novel analytic techniques, possibly leveraging additional structures from additive combinatorics or the latest bounds for exponential sums.
• Transference to Other Algebraic Functions: The methodology may be adaptable to more general polynomial or transcendental forms, with a focus on uncovering how algebraic irrationalities influence global distribution features.
• Refinement through Advances in Exponent Pairs: As improvements in exponent pairs materialize (possibly via advances in decoupling theory or Vinogradov mean value theorems), the exponents in the discrepancy bound could sharpen further, approaching the conjectural random model.

Conclusion

This paper delivers sharp second and fourth moment estimates for exponential sums involving square roots and derives an explicit power-saving upper bound on the discrepancy for sums of $D_k(n)$1 square roots modulo one. The results not only bridge a previously unresolved gap between inhomogeneous approximation and distributional uniformity, but also establish a framework—via smoothing and duality—which can potentially extend to broader classes of analytic number theory problems. The explicit exponents and detailed technical arguments invite further advances, both in the direction of proving the conjectured higher moment bounds and in transferring such techniques to parallel distribution problems for other algebraic ensembles.