The van der Corput property for sums of two squares

Abstract: Let $S_N={1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong Sárközy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $ε>0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.

• The paper establishes a quantitative van der Corput property for sums of two squares, yielding a power-saving bound |A| ≪ N^(7/8+ε) for difference-free subsets.
• It introduces an innovative trivariate exponential sum to construct non-negative weights that effectively control local distribution irregularities modulo q.
• The combined major and minor arc Fourier analysis improves earlier quasipolynomial bounds on Sárkőzy-type sets and opens avenues for further research in additive combinatorics.

Quantitative van der Corput Property for Sums of Two Squares

Introduction and Main Results

The study presented in "The van der Corput property for sums of two squares" (2606.29185) establishes a quantitative power-saving form of the van der Corput property for the set $$S_N = \{1 \leq d \leq N : d = x^2 + y^2 \text{ for some } x, y \in \mathbb{Z}\}$$. This result directly leads to a strong Sárkőzy-type theorem: for any $A \subseteq [N]$ with $(A-A) \cap S_N = \varnothing$, the cardinality satisfies $|A| \ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon > 0$, which improves upon prior quasipolynomial bounds.

The claimed exponent, $7/8+\varepsilon$, is currently optimal as exhibited by Younis's construction of sets $A \subseteq [N]$ with $|A| \gg N^{1/2}$ but $(A - A) \cap S_N = \varnothing$. The authors’ approach is based around the construction of non-negative coefficients $c_d$ supported on $A \subseteq [N]$0 such that $A \subseteq [N]$1 and

$A \subseteq [N]$2

uniformly for all $A \subseteq [N]$3. This quantitative van der Corput bound is then transferred, via standard Fourier analytic techniques, to an upper bound on the size of $A \subseteq [N]$4 avoiding any nonzero differences in $A \subseteq [N]$5.

Technical Innovations

Construction of the Weight and Coefficient System

The main technical contribution is an explicit construction of the weights $A \subseteq [N]$6, reflecting insights obtained from computational experiments with finite models. The authors define a trivariate exponential sum

$A \subseteq [N]$7

where $A \subseteq [N]$8 is a fixed nonnegative smooth cutoff supported in $A \subseteq [N]$9 and $(A-A) \cap S_N = \varnothing$0 is chosen in terms of $(A-A) \cap S_N = \varnothing$1. The normalized trigonometric polynomial, after omitting negligible contributions and proper renormalization, has coefficients $(A-A) \cap S_N = \varnothing$2 supported on $(A-A) \cap S_N = \varnothing$3.

Key features are:

• The use of a third variable $(A-A) \cap S_N = \varnothing$4 (beyond $(A-A) \cap S_N = \varnothing$5) supports better control over local distributions modulo $(A-A) \cap S_N = \varnothing$6, aligning with patterns observed in the linear programming step.
• The resulting coefficients favor $(A-A) \cap S_N = \varnothing$7 with large square divisors, reflecting the combinatorial sparsity of $(A-A) \cap S_N = \varnothing$8.
• The multivariate smooth cutoffs and exponential decay in $(A-A) \cap S_N = \varnothing$9 localize all arithmetic contributions to scales relevant for application of Poisson summation and Weyl differencing.

Major and Minor Arc Analysis

The authors use a classical Hardy-Littlewood circle method dichotomy.

Major Arc Estimates:

For $|A| \ll_\varepsilon N^{7/8+\varepsilon}$0 near rationals $|A| \ll_\varepsilon N^{7/8+\varepsilon}$1 with $|A| \ll_\varepsilon N^{7/8+\varepsilon}$2 and $|A| \ll_\varepsilon N^{7/8+\varepsilon}$3 not too large, Poisson summation is applied to the exponential sum, leading to principal terms governed by complete exponential sums $|A| \ll_\varepsilon N^{7/8+\varepsilon}$4: $|A| \ll_\varepsilon N^{7/8+\varepsilon}$5 Crucially, the paper proves that $|A| \ll_\varepsilon N^{7/8+\varepsilon}$6 always lies in a fixed sector $|A| \ll_\varepsilon N^{7/8+\varepsilon}$7, so its real part is positive and bounded away from zero for all $|A| \ll_\varepsilon N^{7/8+\varepsilon}$8, including problematic moduli like high powers of $|A| \ll_\varepsilon N^{7/8+\varepsilon}$9 or primes $\varepsilon > 0$0.

Minor Arc Estimates:

On arcs away from such rationals, Weyl differencing and second-moment bounds for quadratic exponential sums are deployed, producing exponentially decaying error terms. The controlling parameter is the scale $\varepsilon > 0$1 (of order $\varepsilon > 0$2), and the minor arc “saving” is of size $\varepsilon > 0$3.

The union of both arc analyses leads to the claimed lower bound for $\varepsilon > 0$4, uniformly for all $\varepsilon > 0$5.

Numerical Results and Bounds

The principal numerical statement is the upper bound of $\varepsilon > 0$6 for $\varepsilon > 0$7. This significantly strengthens previously known quasipolynomial bounds, where the best previous decay was $\varepsilon > 0$8. The construction, by Younis, of sets with $\varepsilon > 0$9 having no differences in $7/8+\varepsilon$0 shows that while the exponent $7/8+\varepsilon$1 is not best possible, improvement below $7/8+\varepsilon$2 is unachievable.

Implications and Further Directions

This work deepens the connection between Fourier-analytic “van der Corput property” and arithmetic combinatorics relating to sumsets and difference sets defined by values of structured forms—in this case, the sums of two squares. The passage from exponential sum lower bounds to upper bounds on subset size is standard but powerful, and the power-saving achieved here is a marked advancement over previous approaches, especially those relying on the recurrence properties or density increment among structured sets.

From a methodological perspective, the approach is distinct from recent work on shifted primes and their differences (e.g., Green's power-saving for $7/8+\varepsilon$3 differences), since the set of sums of two squares is much less equidistributed modulo $7/8+\varepsilon$4. The technique of supplementing double-variable exponential sums with an auxiliary $7/8+\varepsilon$5 parameter addresses local irregularities in this distribution.

The numerical improvements in Sárkőzy-type theorems suggest avenues for further tightening the exponent, perhaps by more refined analysis of the underlying exponential sums or leveraging deeper harmonic analytic tools. More generally, analogous van der Corput bounds for other binary quadratic forms or value sets of higher complexity may be accessible by similar means, at the cost of increased technical complexity in modulo $7/8+\varepsilon$6 analysis.

Future developments in AI and computational mathematics may further accelerate the process of discovering, visualizing, and conjecturing optimal trigonometric weights, as evinced by the effective deployment of linear programming experiments in this work.

Conclusion

The paper establishes a quantitative van der Corput property for the sums of two squares, improving the bound in a Sárkőzy-type theorem for sets avoiding such differences to $7/8+\varepsilon$7. The construction and analysis of appropriate trigonometric weights is noteworthy, especially the resolution of local distribution anomalies through variable localization and sector bounds for complete exponential sums. This work sets a template for further quantitative results in the additive combinatorics of sets defined by arithmetic structures and motivates future attempts to further reduce exponents or extend to additional nonlinear value sets.