Bounding the exponential sum on squares of some sifted sequences

Abstract: Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.

• The paper establishes upper bounds for exponential sums over squares in sifted sequences, matching prime-related results up to logarithmic factors.
• It employs advanced sieve methods, bilinear forms, and circle method adaptations to effectively manage quadratic exponential sums.
• The findings extend analytic techniques from prime number theory to sparse Gaussian integer sequences, influencing additive number theory and sieve theory.

Summary and Analysis of "Bounding the exponential sum on squares of some sifted sequences" (2604.09448)

Problem Setting and Mathematical Context

The paper addresses upper bounds for exponential sums taken over the squares of elements in certain "sifted" or arithmetic sequences, particularly focusing on the set $B$ of odd primitive Gaussian integers. Formally, for $\alpha \in \mathbb{R}$ and $N>0$, the exponential sum considered is

$S(\alpha; N) = \sum_{n \leq N} b(n) e(n^2 \alpha),$

where $b(n)$ is the characteristic function of $B$. The set $B$ consists of odd integers expressible as sums of two coprime squares, or equivalently, elements in $\mathbb{Z}[i]$ of norm congruent to $1\ (\mathrm{mod}\ 4)$ with certain factorization restrictions.

Bounding sums of this type is central in analytic number theory, with connections to classical results on primes (e.g., Vinogradov’s proof of the ternary Goldbach conjecture), Waring’s problem, and the distribution of arithmetic sequences in arithmetic progressions or on major/minor arcs in the Hardy-Littlewood circle method.

Main Theorems and Results

Theorem 1: Major Arc Bounds for Sifted Sequences of Squares

Let $|\alpha - a/q| < 1/q^2$ with coprime $\alpha \in \mathbb{R}$0 and $\alpha \in \mathbb{R}$1 the indicator of $\alpha \in \mathbb{R}$2. The paper establishes the uniform bound

$\alpha \in \mathbb{R}$3

for any $\alpha \in \mathbb{R}$4.

This result matches, up to logarithmic factors and $\alpha \in \mathbb{R}$5-powers, the sharpest known bounds for exponential sums over the sequence of all primes squared (as obtained by A. Ghosh), and thus demonstrates that the Gaussian integer sequence $\alpha \in \mathbb{R}$6 exhibits similar pseudorandomness/structural complexity in this analytic context.

Theorem 2: Mean Value Estimates over Multiple Frequencies

For the sum over $\alpha \in \mathbb{R}$7,

$\alpha \in \mathbb{R}$8

an analogous major arc estimate is proven: $\alpha \in \mathbb{R}$9 This is relevant when considering the distribution of $N>0$0 for $N>0$1, or more generally, when integrating these exponential sums over a family of dilations.

Generalizations

The method is robust: it directly extends to sifted sequences defined via other quadratic forms (e.g., primitive Loeschian integers or combinations thereof), providing uniform bounds with the same asymptotic pattern. These cases correspond to higher sieve dimensions, reflecting increased arithmetic restrictions, but the analytic machinery remains applicable.

Techniques and Innovations

The analysis draws on a combination of advanced sieve methods (including combinatorial decompositions adapted from recent work by Ramaré and Viswanadham), Weyl-van der Corput and Bombieri–Vinogradov type inequalities, and deep results on the rational approximation of quadratic forms (notably invoking works of Browning and Heath-Brown for controlling the number of solutions to certain quadratic Diophantine equations).

A crucial structural innovation is the adaptation of the circle method and bilinear forms technology—tools generally specialized for primes or full squares—into the context of these highly sifted sets. This involves careful coprimality management and nontrivial manipulations involving Dirichlet characters to handle congruence and square conditions, as well as precise invocations of Vinogradov-type lemmas for controlling exponential sums with rational approximations.

Strongest Numerical Claims

• The bound $N>0$2 is of the strength previously known only for primes and their squares, holding for the considerably thinner set $N>0$3.
• The same analytic machinery yields powerful mean value results uniform in $N>0$4.

No instances are provided where strict improvements upon prior bounds for primes or Weyl sums are obtained, but the significant contribution is the extension and unification of these bounds for much more general sifted sequences.

Implications and Future Directions

Theoretical Implications

This work demonstrates that sophisticated analytic and algebraic techniques used for the primes, especially bilinear forms and the Vinogradov–Vaughan method, are adaptable to a far broader class of sequences, with implications for understanding the distribution of values of quadratic polynomials on general sifted sets.

These results facilitate further analysis of distribution properties (e.g., equidistribution, local statistics, value distribution of quadratic forms) for arithmetic sequences with intricate structural conditions, bridging random-like behavior and deep algebraic symmetry.

Practical and Broader Impact

Several applications can be foreseen:

• Additive Number Theory: The bounds directly impact the analysis of representations in sums of squares, the study of Waring’s type problems for thin sequences, and density results for special quadratic forms.
• Sieve Theory: The work suggests that spectral methods and bilinear sum decompositions—a staple for primes—can be extended to higher-dimensional sieves and more complex forms, potentially influencing algorithms for computational number theory.
• Analytic Techniques: The established techniques may catalyze further advances in bounding exponential sums with polynomial arguments in other structured arithmetic sets, enriching the toolkit for dealing with major and minor arc problems in the circle method.

Prospective Developments

• Extension to Higher Degree Forms and Sieve Dimensions: Future research may consider whether similar approaches provide nontrivial bounds for exponential sums over even thinner or more highly structured sets, including thin orbits of arithmetic groups, or in contexts with non-diagonal forms.
• Applications to Distribution of Arithmetic Functions: The improved control granted by these bounds may facilitate new results on the autocorrelation of arithmetic functions along quadratic (or higher) sequences.
• Refinements and Optimality: Further work may address whether the $N>0$5-loss and logarithmic denominators are optimal, or if sharper methods can eliminate these factors in special cases.

Conclusion

This paper rigorously demonstrates that upper bounds on exponential sums over squares—previously proven for highly structured sequences like primes—can be matched for considerably sparser sifted sets, such as odd primitive Gaussian or Loeschian integers. Through careful application of bilinear sum analysis, Vinogradov’s lemma, and results on quadratic Diophantine equations, it establishes strong, nearly optimal major arc bounds and mean value estimates. This greatly expands the analytic understanding and potential reach of exponential sum estimates in arithmetic sequences, providing both new tools and new perspectives for ongoing research in analytic number theory.