Gaps in Multiplicative Sidon Sets

Abstract: For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq{1,2,\dots,n}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $ρ= \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{ρ+\varepsilon}$ for every $\varepsilon > 0$.

• The paper establishes that every multiplicative Sidon set in {1,…,n} intersects all intervals of length ⌊√n⌋, affirmatively resolving Sárközy's long-standing question.
• It combines elementary combinatorial constructions with analytic number theory, using prime gap estimates to derive a gap bound with an exponent of approximately 0.46934.
• The work employs AI-assisted formal verification in Lean, illustrating a breakthrough in integrating automated methods with traditional mathematical proof techniques.

Gaps in Multiplicative Sidon Sets: Formal Results and Asymptotics

Introduction

The study of multiplicative Sidon sets, i.e., integer sets $A$ such that the products $aa'$ with $a, a' \in A$ and $a \leq a'$ are all distinct, is a foundational topic at the intersection of additive number theory and combinatorics. While the maximal cardinality of such sets in $\{1,2,\ldots, n\}$ is asymptotically well understood, much less was known about how "evenly" these sets can be distributed—in particular, about the maximal gap between consecutive elements. The paper "Gaps in Multiplicative Sidon Sets" (2605.02064) delivers the first unconditional, nontrivial upper bounds that essentially match known lower bounds, introducing new methods in combinatorial construction, analytic number theory, and formal theorem verification.

Problem Definition and Context

Let $g(n)$ denote the infimum of real $L$ such that there exists a multiplicative Sidon set $A \subseteq \{1, ..., n\}$ intersecting every real interval $[x, x+L] \subseteq [1, n]$. In 2001, Sárközy posed the question whether $g(n) \leq \sqrt{n}$ for all $aa'$0 and whether significantly smaller maximal gaps are attainable. As the set of primes is multiplicative Sidon, gaps are inextricably linked to classical prime gap estimates. However, even the Riemann Hypothesis does not provide the necessary prime gap bounds to settle Sárközy's question directly.

Main Theorems and Proof Techniques

The $aa'$1 Bound

The authors establish that $aa'$2 for all $aa'$3, resolving Sárközy's question affirmatively and unconditionally. The construction is explicit and elementary: given $aa'$4, the set $aa'$5 is shown to be multiplicative Sidon and to intersect all intervals of length $aa'$6. This is a nontrivial claim, requiring the demonstration that $aa'$7 implies $aa'$8 and $aa'$9, which is achieved through a concise algebraic argument. Notably, this proof was autonomously discovered and then formally verified in Lean by Aristotle, signifying a methodological advancement in the integration of AI-aided formal mathematics.

Sub-Power Bounds via Analytic Number Theory

The paper goes further, leveraging the Baker-Harman-Pintz bound on prime gaps and a technical result of Laishram–Murty to show that for any $a, a' \in A$0, there exists $a, a' \in A$1 such that

$a, a' \in A$2

This advances the exponent from the best known achievable via prime gaps ($a, a' \in A$3) into a regime below $a, a' \in A$4. The proof combines sophisticated weighting in a bipartite matching argument (a weighted variant of Hall's theorem) with careful use of primes represented as $a, a' \in A$5 in short intervals, parameter optimization, and summation bounds extracted from [Laishram–Murty, 2012].

Numerical Results and Strength of Bounds

Key numerical implications are:

• The construction ensures every interval of length $a, a' \in A$6 within $a, a' \in A$7 contains an element of a multiplicative Sidon set.
• The exponent $a, a' \in A$8 in the power-saving bound is strictly lower than $a, a' \in A$9, establishing gaps smaller than $a \leq a'$0.
• The lower bound $a \leq a'$1 follows trivially from density considerations.
• The result is established for all sufficiently large $a \leq a'$2, as the construction covers small values by a finite (adjustable) constant.

Formal Verification and AI Integration

A distinctive aspect of this work is the rigorous use of formal verification. The proof of the $a \leq a'$3 gap bound is not only humanly elementary but also brief enough to be independently discovered by an automated theorem prover (Aristotle), and then machine-checked in Lean. The more technical proof for the refined exponent is also formalized. This signals a practical shift where high-level research in combinatorial number theory can be both assisted and certified by AI tools, bolstering assurance in correctness, especially in arguments orchestrating complex combinatorial and analytic constraints.

Theoretical and Practical Implications

The findings illuminate the structure of multiplicative Sidon sets beyond mere extremal cardinality. They establish that large Sidon sets can be constructed with uniformly small gaps, pushing the theory closer to our understanding in the additive analogue. Practically, the methodology is relevant for any application where tuples with multiplicatively independent elements are required, and where regular coverage is advantageous—e.g., cryptography, random coding, and pseudorandom number generation.

An even more profound implication is the demonstration of effective, large-scale Lean formalization of nontrivial analytic-combinatorial theorems, particularly via autonomous AI agents. This signals that future research in analytic number theory might increasingly rely on automated discovery and verification, especially for extremal problems with delicate parameter choices.

Future Directions

There remain several clear avenues for extension:

• Improving the exponent $a \leq a'$4 in the sub-$a \leq a'$5 power bound. This would likely require new breakthroughs in the distribution of primes, e.g., sharper short-interval results.
• Investigating precise lower bounds for $a \leq a'$6: for instance, true lower bounds greater than $a \leq a'$7 are not excluded.
• Seeking constructions with additional structure or constraints (e.g., additive-multiplicative Sidon sets).
• Adapting the AI-based formal discovery pipeline to related extremal combinatorial questions.

Conclusion

"Gaps in Multiplicative Sidon Sets" (2605.02064) establishes that, for any $a \leq a'$8, one can construct a multiplicative Sidon set in $a \leq a'$9 intersecting every interval of length at most $\{1,2,\ldots, n\}$0, with consequent improvements showing gaps as small as $\{1,2,\ldots, n\}$1. These arguments combine explicit combinatorial constructions, deep analytic input, and AI-powered formal verification, collectively resolving a long-standing problem and setting new paradigms for research at the combinatorics–number theory interface.