The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations (2501.10203v1)

Abstract: A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on binary systems of linear forms and of Kelley and Meka on Roth's theorem on arithmetic progressions, we show that, for $N \geq \exp((k\log(2/\alpha))^{O(1)})$, any subset $A \subseteq [N]$ of density at least $\alpha$ contains a $k$-configuration. This improves on the previously best known bound $N \geq \exp((2/\alpha)^{O(k^2)})$, due to Shao. As a consequence, it follows that any finite non-empty set $A \subseteq \mathbb{Z}$ contains a subset $B \subseteq A$ of size at least $(\log|A|)^{1+\Omega(1)}$ such that $b_1+b_2 \not\in A$ for any distinct $b_1,b_2 \in B$. This provides a new proof of a lower bound for the Erd\H{o}s--Moser sum-free set problem of the same shape as the best known bound, established by Sanders.

• The paper sharpens bounds on k-configurations, enabling the identification of larger sum-free subsets in integer sets.
• It leverages improved density and structural arguments to replace previous exponential dependencies with more efficient polynomial bounds.
• The work applies modern combinatorial methods to deepen our understanding of additive number theory and the behavior of sum-free sets.

An Analytical Overview of the Erdős–Moser Sum-Free Set Problem

The paper "The Erdős–Moser sum-free set problem via improved bounds for $k$-configurations" by Adrian Beker presents significant results in additive combinatorics, focusing on sum-free sets within integer sets, motivated by a question posed by Erdős and Moser. The problem inquires about the largest subset $B$ of a given integer set $A$ where the subset $B$ is sum-free concerning $A$, meaning no two elements in $B$ sum to an element in $A$.

To address this question, the author discusses the concept of $k$-configurations, which are sets of integers with pairwise arithmetic means also included in the set. Recent advances by researchers like Kelley and Meka, and Filmus et al., on three-term arithmetic progressions and linear forms, respectively, are harnessed. Building on these foundations, Beker establishes an improved bound for finding $k$-configurations in subsets of integers, refining Shao's previous results. For subsets of integers with density at least $\alpha$, the paper shows that for a large $N$, any subset $A \subseteq [N]$ contains a $k$-configuration. The bounds given are $N \geq \exp((k \log(2/\alpha))^{O(1)})$, improving from the former $N \geq \exp((2/\alpha)^{O(k^2)})$.

Using these analytical improvements and the tools from additive combinatorics, the paper provides a new proof of the lower bound for the Erdős–Moser problem with a shape similar to Sanders' best-known result. Specifically, any finite non-empty set $A \subseteq \mathbb{Z}$ includes a sum-free subset $B$ with a size growing super-logarithmically with respect to $|A|$.

Key Numerical and Theoretical Insights

1. Improved Bounds: The pivotal result improves the bound for $k$-free configurations, substituting previous exponential complexity dependencies on $k^2$ with polynomial ones, immensely strengthening the results for large $k$-configurations.
2. Density and Configuration Interplay: The novel bound depends significantly on the product $k^{68} L(\alpha)^{16}$. Exploring the dependence on both the density and the configuration is addressed through a combination of structural and density increment insights.
3. Applications in Combinatorial Number Theory: The results have deeper implications for determining the structure of subsets within number sets, contributing to our understanding of density theorems and sum-free sets.
4. Technical Advances in the Proof: The paper implements Kelley–Meka techniques in a new context, specifically the unique setting of systems of linear forms of pairs rather than individual elements, showcasing the versatility of these methods in proving density increments.

Implications and Future Directions

Practically, the improved bounds for $k$-configurations extend the ability to characterize large sparse sets within number theory and related applications. The extension to arbitrary coefficients would likely require further paper but promises more general applicability.

Theoretically, it underscores how the complexity of finding certain configurations in sets can be reduced, paving the way for more efficient methods in solving related problems, such as understanding three-term arithmetic progressions in subsets of integers. Additionally, the conjecture presented indirectly suggests that examining the nature of density increments further could unveil more efficient ways of breaking through existing barriers in the field of additive combinatorics.

Conclusion

Beker’s work refines existing theorems and integrates modern combinatorial techniques to not only answer a historical question posed by Erdős and Moser more robustly but also lays the groundwork for future inquiries into configurations' behavior in arithmetic settings. This improvement in bounds is expected to have broad implications across various domains where counting configurations within sets is a fundamental challenge, thus contributing significantly to the broader landscape of mathematical research in the context of sum-free sets and configuration problems.