[Paper Review] Correlation estimates for sums of three cubes

Joerg Bruedern, Trevor D. Wooley|Bristol Research (University of Bristol)|Apr 15, 2014

Limits and Structures in Graph Theory13 references3 citations

TL;DR

This paper establishes improved correlation estimates for sums of three positive integral cubes using the Hardy-Littlewood circle method, proving that systems of homogeneous linear equations in sums of three cubes are soluble whenever the number of variables exceeds twice the number of equations. The key result is a sharp bound of $\ll N^{r+1/6+\varepsilon}$ on correlation sums, significantly improving upon classical estimates when $r > 1$. The analysis leverages smooth number restrictions and minor arc estimates to control representation counts and ensure positivity of solutions in the required forms.

ABSTRACT

We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of systems of homogeneous linear equations in sums of three positive cubes whenever these systems have more than twice as many variables as equations.

Motivation & Objective

• To establish improved upper bounds for linear correlation sums involving sums of three positive integral cubes.
• To analyze the solubility of systems of homogeneous linear equations in sums of three positive cubes.
• To extend classical correlation estimates beyond the folklore bound $N^{7r/6+\varepsilon}$ using smooth number restrictions and minor arc analysis.
• To demonstrate that systems with more than twice as many variables as equations are soluble in sums of three positive cubes under general position conditions.

Proposed method

• Utilizes the Hardy-Littlewood circle method to analyze exponential sums associated with sums of three cubes.
• Introduces a mollified version $\rho_\eta(n)$ of the representation function $\rho(n)$, restricting representations to those with smooth variables.
• Applies Hua's lemma and Weyl's inequality to control major and minor arc contributions in the circle method.
• Employs the concept of highly non-singular matrices to ensure linear independence of associated linear forms.
• Uses Cauchy-Schwarz and symmetry to bound correlation sums via $\ell^2$-norm estimates of $\rho_\eta(n)$.
• Applies minor arc estimates with bounds of the form $\ll P^{3s-3r-\delta}$ to eliminate spurious solutions and isolate main terms.

Experimental results

Research questions

• RQ1 Can correlation sums involving sums of three positive cubes be bounded more sharply than the classical $N^{7r/6+\varepsilon}$ estimate for $r > 1$?
• RQ2 Under what conditions is a system of homogeneous linear equations in sums of three cubes guaranteed to have a solution in positive integers?
• RQ3 Can the use of smooth number restrictions improve the error term in correlation sum estimates for sums of three cubes?
• RQ4 What is the minimal number of variables required for solubility of such systems, and how does this relate to the number of equations?
• RQ5 How do minor arc estimates and representation counts interact to ensure positivity of solutions in the required forms?

Key findings

• The correlation sum $\Xi_{2r}(N;A;\mathbf{h})$ is bounded by $\ll N^{r+1/6+\varepsilon}$ for highly non-singular $A \in \mathbb{Z}^{r \times 2r}$, improving upon the classical $N^{7r/6+\varepsilon}$ bound for $r > 1$.
• For the mollified representation function $\rho_\eta(n)$, the correlation sum $\Xi_{2r,\eta}(N;A;\mathbf{h})$ satisfies $\ll N^{r+\xi+\varepsilon}$ with $\xi = (\sqrt{2833} - 43)/123 < 1/12$, under suitable $\eta$.
• Systems of $r$ homogeneous linear equations in $s > 2r$ variables are soluble in sums of three positive cubes whenever the coefficient matrix is highly non-singular and a real solution exists in positive reals.
• The minor arc contribution is bounded by $\ll P^{3s-3r-\delta}$ for some $\delta > 0$, ensuring the main term dominates.
• The number of solutions $\Upsilon(N)$ to such systems satisfies $\gg N^{s-r}$, after removing high-multiplicity solutions via smooth number filtering.
• The proof establishes that the number of solutions with high representation counts is negligible, ensuring the main term is not overwhelmed by multiplicity.