A combinatorial large sieve for Sidon sets, distances, and norm forms

Abstract: We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset{1<sup>2,\ldots,N<sup>2}$ satisfies [ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) ] for some absolute constant $c&gt;0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]<sup>2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a super-polylogarithmic saving for subsets of $[N]<sup>2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. Moreover, we prove the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

• The paper introduces a novel combinatorial large sieve that delivers super-polylogarithmic upper bounds for Sidon sets and norm forms.
• It leverages modular splitting and entropy techniques to refine classical estimates in additive combinatorics, particularly in the grid distance problem.
• The method extends to B_h[g]-sets and norm form representations in number fields, establishing new thresholds for extremal combinatorial problems.

Combinatorial Large Sieve Methods for Bounded-Multiplicity Sets: Sidon Sets, Distances, and Norm Forms

Introduction and Context

The paper "A combinatorial large sieve for Sidon sets, distances, and norm forms" (2606.17487) introduces a new paradigm for sieve methods in additive combinatorics, specifically tailored to sets with bounded algebraic multiplicities. The approach unifies tools from the analytic large sieve with entropy-based combinatorial principles to achieve strong upper bounds on the sizes of diverse families such as Sidon sets, sets with distinct distances, and sets controlling the representation of norm forms in number fields. The technical innovations arise from exploiting algebraic splitting in congruences modulo many small primes and chaining this with entropy deficit arguments or direct combinatorial weightings.

Main Results and Theoretical Advances

Sidon Sets and Ill-Distributed Families

For Sidon sets in intervals and their analogues in the set of squares, the combinatorial large sieve provides the first super-polylogarithmic saving in upper bounds over classical results. Specifically, for any Sidon set $A \subset \{1^2, 2^2, \ldots, N^2\}$, the authors show

$$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$

for some $c > 0$. This substantially improves upon the upper bound of $O(N / (\log N)^{1/2})$ of Hanson for Sidon sets in the squares, making significant progress on a problem of Erdős and Alon. This is achieved via a combinatorial large-sieve lemma that leverages how the perfect squares are ill-distributed modulo small primes, and how the Sidon set property links to difference bounds.

Grid Distance Problems

The method generalizes to geometric settings. Let $g(N)$ be the maximal size of a subset of $[N]^2$ with all pairwise Euclidean distances distinct, i.e., with no repeated distance. The authors obtain

$$g(N) \lesssim N \exp\left(-c \frac{\log N}{\log \log N}\right),$$

constituting the first reduction of the exponent below $N$ in more than three decades for this Erdős-Guy problem. Similarly, for the maximal size $f(N)$ of a subset of $[N]^2$ with no isosceles triangle (extending results relevant to the so-called PatternBoost and MathOverflow contexts), they achieve

$$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$0

Both results rely on the insight that, modulo appropriate primes, quadratic forms such as $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$1 split into lines, creating numerous modular coincidences unless the set is highly restricted—a situation that is precisely controlled by the combinatorial sieve with bounded-multiplicity hypotheses.

Entropic Sieve and Generalized Sidon Sets

For sets $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$2 solving higher-order representation equations with bounded multiplicity (so-called $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$3 sets), the authors develop an entropy-enhanced version of the large sieve. This new approach goes beyond the classical spread-vs-entropy dichotomy by using weight functions and Shearer's inequality to realize more of the potential congruence-based savings:

• For $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$4-sets in the squares, an optimal bound of the form

$$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$5

is established.

• For $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$6-sets in cubes and $$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$7-sets in fourth powers (the first nontrivial bounds for these problems), they obtain strong sublinear-in-$$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$8 savings, e.g., for cubes,

$$|A| \leq N \exp\left(-c \frac{\log N}{\log \log N}\right)$$9

under suitable growth conditions on $c > 0$0.

The entropy-based approach is robust and applies whenever sufficient algebraic factorization occurs modulo many small primes.

Norm Form Extensions

The authors generalize their framework to the context of ring of integers of number fields $c > 0$1, bounding the size of finite sets $c > 0$2 with bounded norm form representations. If $c > 0$3 satisfies a bound of the form

$c > 0$4

for the representation counting function associated to norm forms, they prove

$c > 0$5

where $c > 0$6. When specialized to boxes in $c > 0$7, this yields bounds of the same shape for solutions to norm form equations with bounded multiplicity, reproducing and extending the results for quadratic forms to higher-degree settings.

Methodology

The strategy systematically integrates combinatorial, analytic, and information-theoretic principles:

• Combinatorial Large Sieve: Refinement of classical large/larger sieve arguments, with control retained through the explicit tracking and bounding of modular differences via Cauchy-Schwarz and direct counting.
• Algebraic Branching Modulo Primes: Key use of the fact that, for suitable algebraic forms (squares, quadratic forms, norm forms), the solution space often splits into many subspaces or lines modulo good primes.
• Entropy Techniques: Application of Shearer's inequality to extract entropy deficits from the lack of uniformity in residue class distributions, yielding global savings in set size constraints. A weighted variant is also introduced by combining local modular weights to construct global weight functions that admit both upper and lower bounds in terms of entropy and the multiplicative structure of the forms.
• Arithmetic Inputs: Use of explicit counting for algebraic equations over finite fields—such as the number of solutions to $c > 0$8 in $c > 0$9, or for norm equations over rings of algebraic integers.

Numerical Strength and Contradictory Claims

The paper's bounds represent the first super-polylogarithmic improvements for Sidon sets in the squares and for distinct-distance and isosceles-free sets in the integer grid. Notably, for Sidon subsets of $O(N / (\log N)^{1/2})$0, the upper bound now includes an exponential in $O(N / (\log N)^{1/2})$1, which was previously unattainable with available methods.

A contradictory aspect to prior folklore is the successful extension of large sieve principles—traditionally analytic and uniformity-based—into nonuniform, sparse algebraic domains by leveraging bounded-multiplicity in representations. The entropic enhancement further breaks the conventional intuition that sieve savings in these settings are capped at polylogarithmic factors.

Implications and Future Directions

The combinatorial large sieve and its entropy-weighted variants furnish new upper bounds for longstanding extremal problems and establish a template for analyzing sets with additive or geometric constraints in high-dimensional and algebraic settings. The methods connect to inverse sieve problems, showing that dense sets with nontrivial modular structure are either small or highly structured—strengthening ongoing work on the inverse sieve conjectures.

On the theoretical side, the findings suggest that algebraic splitting modulo primes, together with bounded-multiplicity, can often drive extremal upper bounds far below what would be predicted by traditional heuristics. On the practical side, these advances narrow the possible behavior of combinatorially rich sets, setting new milestones for what constructions are possible.

For future developments:

• The extension to higher $O(N / (\log N)^{1/2})$2 in $O(N / (\log N)^{1/2})$3 sets, especially for perfect powers, appears to be fundamentally restricted by arithmetic properties (e.g., for $O(N / (\log N)^{1/2})$4), as the required algebraic splitting does not provide sufficient gain.
• There are possible cross-applications with recent number-field based constructions for additive and geometric extremal problems.
• The entropy-based sieve and its weighted form are potentially adaptable to other settings where algebraic structure and modular splitting coincide, e.g., sum-product phenomena and design matrices.

Conclusion

This work delivers the most substantial progress to date on Sidon-type and distance-avoidance problems in combinatorics with bounded-multiplicity constraints by introducing a combinatorial large sieve method and its entropy-enhanced forms. The integration of modular splitting, combinatorial weights, and entropy constitutes a versatile framework with applications across additive number theory, discrete geometry, and arithmetic combinatorics, setting new standards for upper bound estimations in these domains.