Gaps in Multiplicative Sidon Sets II

Abstract: With $ρ= \frac{13-\sqrt{69}}{10} \approx 0.47$, it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = cd$) in ${1, 2, \ldots, n}$ with maximal gap size $\ll_{\varepsilon} n^{ρ+ \varepsilon}$. Here we improve upon this result and show that one can take $ρ= \frac{10}{33} \approx 0.303$ instead.

• The paper establishes a new upper bound exponent of 10/33 for maximal gaps in multiplicative Sidon sets on [n].
• It synthesizes probabilistic methods, combinatorial techniques, and refined prime distribution estimates to significantly improve previous bounds.
• The approach leverages formal verification in Lean and advanced analysis of exceptional prime intervals to enhance understanding of Sidon set structure.

Improved Upper Bounds on Gaps in Multiplicative Sidon Sets

Overview

The paper "Gaps in Multiplicative Sidon Sets II" (2606.07428) presents new, stronger upper bounds for the maximal gaps in multiplicative Sidon subsets of $[n] = \{1,2,\ldots,n\}$, improving previous work that established an upper bound exponent $\rho \approx 0.47$ to a new value $\rho=10/33\approx 0.303$. The authors achieve this by integrating probabilistic methods, advanced estimates on the distribution of primes in short intervals, and extremal combinatorial reasoning, with several components formalized in Lean and leveraging automated theorem proving.

Mathematical Framework

Multiplicative Sidon sets are defined as subsets $A\subset\mathbb{N}$ such that any equation $ab=cd$ with $a,b,c,d\in A$ implies $\{a,b\}=\{c,d\}$. The classical set of primes is multiplicative Sidon, but the theory pursues much larger constructions and seeks control over their density and, critically, the maximal "gap" between elements. The function $g(n)$, the infimum over all $L$ for which there exists a multiplicative Sidon subset of $[n]$ intersecting every interval of length $\rho \approx 0.47$0, forms the central object of study.

Previous major advances established $\rho \approx 0.47$1 with $\rho \approx 0.47$2 ("$\rho \approx 0.47$3"), recently improved to $\rho \approx 0.47$4 via probabilistic and graph-theoretic constructions. The present work targets the full range down to conjectural limits suggested by prime distribution hypotheses, aiming for the tightest obtainable unconditional bounds.

Main Results

The core technical result is:

• Theorem (Main): There exists a multiplicative Sidon subset of $\rho \approx 0.47$5 with maximal gap $\rho \approx 0.47$6.

This is a significant improvement, advancing closer to the conjectured behavior if one assumes strong unproven assertions about prime gaps.

The results are articulated through two layers of innovation:

1. Probabilistic Construction with Local Lemma: By partitioning $\rho \approx 0.47$7 into intervals and selecting elements using the asymmetric Lovász Local Lemma, the existence of multiplicative Sidon sets intersecting all intervals of length $\rho \approx 0.47$8 is established without recourse to deeper analytic number theory. This method quantifies combinatorially rare "bad" events where multiplicative Sidonness would be violated.
2. Refined Application Exploiting Prime Distribution: By leveraging recent fine-grained analyses of "prime-poor intervals"—regions where the count of primes is abnormally low—the construction is refined. Key here are results of Gafni and Tao (Gafni et al., 29 May 2025), which tightly bound the measure of exceptional intervals where the prime number theorem badly fails for intervals of the form $\rho \approx 0.47$9. These results allow for a significantly lower bound on the size of exceptional intervals and, by careful combinatorial/probabilistic selection, yield multiplicative Sidon sets with much smaller maximal gap. This yields the exponent $\rho=10/33\approx 0.303$0 in the upper bound.

Analytical Innovations

The argument is structured around a meticulous interplay between prime distribution estimates and combinatorial selection. The notion $\rho=10/33\approx 0.303$1 is introduced to quantify the size of the exceptional set of "prime-poor" intervals, and the critical criterion derived is that if $\rho=10/33\approx 0.303$2, then $\rho=10/33\approx 0.303$3.

Two major analytical tools are involved:

• Lovász Local Lemma (LLL): Used to guarantee, with positive probability, that a random selection from disjoint blocks avoids all combinatorial configurations leading to non-trivial $\rho=10/33\approx 0.303$4 equalities.
• Estimates for Prime-Poor Intervals: Specific effective bounds for the measure of intervals failing to contain "enough" primes are gleaned from recent literature. The main new numeric improvement draws from detailed results in [Gafni, Tao 2025].

The connection between exceptional sets for the prime number theorem and Sidon set construction is deepened, with explicit dependence shown: any new progress on the measure of exceptional prime intervals directly translates to sharper bounds for $\rho=10/33\approx 0.303$5.

Implications and Prospects

The strong explicit upper bound $\rho=10/33\approx 0.303$6 offers a new benchmark for extremal constructions of multiplicative Sidon sets. From a practical standpoint, these methods provide a framework for constructing dense Sidon sets with controlled maximal gaps, relevant to combinatorial number theory, additive combinatorics, and applications such as cryptography where "uniqueness of product" properties are valued.

Theoretically, the results highlight the interplay between number theory and probabilistic combinatorial techniques. The approach demonstrates that improvements in our understanding of the primes in short intervals (including via analytic or sieve-theoretic advances) have direct combinatorial consequences. Furthermore, the formalization using automated theorem proving tools (Lean, Aristotle) and the integration of methods produced by LLMs signifies the increasing synthesis between mathematical theory and AI-driven formal reasoning.

If conjectures about primes (such as the Generalized Riemann Hypothesis or the Lindelöf Hypothesis) are eventually confirmed, the allowable exponent could, conditional on these conjectures, be reduced to $\rho=10/33\approx 0.303$7, as explained from the consequences for exceptional sets of prime short intervals.

Conclusion

This work advances the upper bound for maximal gaps in multiplicative Sidon subsets of $\rho=10/33\approx 0.303$8 to $\rho=10/33\approx 0.303$9, leveraging advanced probabilistic, analytic, and computational techniques. The results not only improve quantitative bounds but also clarify the dependency of Sidon set structure on the fine-scale distribution of primes. The paper establishes a template for future progress: advancements in the understanding of exceptional sets for the prime number theorem in short intervals can be expected to yield corresponding improvements in the density and gap structure of multiplicative Sidon sets.