An Erdős problem on random subset sums in finite abelian groups

Abstract: Let $f(N)$ denote the least integer $k$ such that, if $G$ is an abelian group of order $N$ and $A \subseteq G$ is a uniformly random $k$-element subset, then with probability at least $\tfrac12$ the subset-sum set ${ \sum_{x \in S} x : S \subseteq A }$ equals $G$. In 1965, Erdős and Rényi proved that for all $N$, $$ f(N) \le \log_2 N + \left(\frac{1}{\log 2}+o(1)\right)\log\log N. $$ Erdős later conjectured that this bound cannot be improved to $f(N)\le \log_2 N+o(\log\log N)$. In this paper we confirm this conjecture by showing that, for primes $p$, $$ f(p)\ge \log_2 p+\left(\frac{1}{2\log 2}+o(1)\right)\log\log p. $$ This work is an outcome of human--AI collaboration: the original qualitative proof was generated autonomously by ChatGPT-5.2 Pro, while the quantitative refinement was developed by the authors.

• The paper provides a counterexample to Erdős's conjecture by proving that in cyclic groups of prime order, the subset-sum threshold necessarily includes an unavoidable log log term.
• The paper employs probabilistic methods, particularly second moment analysis and Poisson heuristics, to derive a sharp lower bound on the subset-sum threshold function f(p).
• The paper’s findings have practical implications for random constructions in cryptography and coding theory, emphasizing the critical influence of group structure on subset sum representations.

A Counterexample to the Erdős Subset-Sum Threshold in Finite Abelian Groups

Introduction

The paper addresses a classical problem in probabilistic group theory originally posed by Erdős, concerning the threshold at which randomly chosen subsets of finite abelian groups almost surely represent all group elements via their subset sums. Let $f(N)$ denote the smallest integer $k$ such that, for every finite abelian group $G$ of order $N$, a uniformly random $k$-element subset $A\subseteq G$ has subset-sum set $\Sigma(A)=\{\sum_{x\in S} x : S\subseteq A\}$ equal to $G$ with probability at least $1/2$. Previous work by Erdős and Rényi established an upper bound of $f(N)\le \log_2 N + O(\log\log N)$, prompting Erdős's conjecture that this $O(\log\log N)$ additive term cannot, in general, be improved to $o(\log\log N)$. This paper provides a sharp counterexample to this conjecture, focused on the case where $G$ is cyclic of prime order, thereby negating the possibility of uniform improvement for all group orders.

Problem Statement and Previous Results

The principal open question, designated Erdős Problem #543, asks whether $f(N)\le \log_2 N + o(\log\log N)$ holds for all $N$. The subset sum problem generalizes familiar combinatorial phenomena, such as random generation in abelian groups, and is closely related to the coin-weighing and zero-sum subset problems. Previous lower bounds for $f(N)$ have been weak: Erdős and Hall showed the upper bound could not universally be improved to $f(N) \le \log_2 N + o(\log\log\log N)$, yet the order of the additive term remained open. The paper confirms that for prime $p$, the error term cannot be reduced to $o(\log\log N)$. Specifically, it is proven that for sufficiently large prime $p$,

$$f(p) \ge \log_2 p + \left(\frac{1}{2\log 2} + o(1)\right)\log\log p.$$

This is strictly larger, by a factor of $1/2$, than the leading second-order term in the upper bound established by Erdős and Rényi, for which the corresponding coefficient is $1/{\log 2}$.

Proof Outline and Main Technical Ingredients

The argument is split into three main components:

1. Reduction to the I.I.D. Model: The proof first shows that the set model (choosing uniformly random $k$-element subsets) is equivalent up to $o(1)$ error to choosing $k$ independent uniformly random elements (the i.i.d. model), due to the small probability of repeated elements when $k = O(\log p)$.
2. Second Moment Analysis and Poisson Heuristics: The main probabilistic estimate concerns the number $U$ of elements of $G$ not attained as a subset sum. For each $x\in G\setminus\{0\}$, the indicator that $x$ is missed by all non-empty subset sums is approximately Poisson distributed with parameter $\lambda\sim (\log p)^\alpha$, $\alpha = c\log 2$ for fixed $c\in (0,1/(2\log 2))$. Experiments with factorial moments and an explicit inclusion-exclusion (Bonferroni) analysis show that, as $p\to\infty$, the expected number $EU$ of missed elements diverges and variance is asymptotically negligible. This ensures, by Chebyshev, that with high probability $U>0$ so $A$ does not generate $G$ via subset sums.
3. Structural Estimation: The technical core involves bounding the contribution to the inclusion-exclusion formula from patterns of subset sums that are not independent due to linear dependencies (low-rank incidence matrices). The lemmas establish that the contribution from low-rank cases is exponentially small compared to the main term. The required bounds on the number of incidence patterns are proven via combinatorial and linear-algebraic arguments (using the stability of rank over $\mathbb{Q}$ and $\mathbb{F}_p$).

Main Result and Numerical Parameters

The central theorem demonstrates that for $k = \lfloor \log_2 p + c\log\log p \rfloor$ with any fixed $c < 1/(2\log 2)$,

$$\mathbb{P}(\Sigma(A) = G) \longrightarrow 0 \quad \text{as} \quad p \to \infty.$$

Hence, $$f(p)\ge \log_2 p + (\frac{1}{2\log 2}+o(1))\log\log p$$.

The constants are quantitatively sharp up to the factor $1/(2\log 2)$, as matching upper bounds are known (with a constant twice as large in the leading order). Achieving the full interval $$f(N)\le \log_2 N + \frac{1}{\log 2}\log\log N + O(1)$$ for all $N$ remains open.

Theoretical and Practical Implications

This result settles Erdős's question (Problem #543) in the negative: the $O(\log\log N)$ additive constant in the threshold for subset sums cannot, in general, be improved for all finite abelian groups. It also characterizes the probability threshold more precisely, thereby influencing our understanding of random covering phenomena in group theory and additive combinatorics.

From a theoretical perspective, the proof exemplifies delicate management of dependencies in probabilistic combinatorics, especially in situations where group structure induces constraints on sumsets. The methods connect subset sum distributions with Poisson local weak limits and employ a refined second-moment approach, with explicit error control over possible incidences.

The paper also highlights, both in its declaration and methodological transparency, the substantial use of AI-assisted theorem discovery and proof refinement. The initial counterexample and most of the conceptual proof outline were generated with a recent LLM; the final step required quantitative sharpening and rigorous validation.

Practically, these results delimit what can be expected from random constructions in cryptography and coding theory (e.g., random generation or representation in finite fields, knapsack-based primitives), especially in regimes where group order and subset sizes scale logarithmically.

Open Directions and Future Work

Several questions remain open. Notably, the exact second-order constant in $f(N)$ is likely strictly larger than the established lower bound, with heuristics and experiments suggesting the value should be $1/\log 2$. Achieving tighter bounds would require a finer analysis of correlation structure among subset sums and improved estimates on the frequency and impact of linear dependencies beyond the current low-rank exclusion machinery.

Furthermore, for particular classes of groups (e.g., elementary abelian $2$-groups), smaller subset sizes suffice, indicating that group structure matters significantly for subset sum coverage. Extending the analysis to more general non-cyclic or composite order groups, or lifting the methods to non-abelian contexts, could yield deeper insight.

Finally, the clear success of automated reasoning and AI in both conceptualizing and formalizing these results paves the way for increased machine involvement in probabilistic combinatorics, perhaps enabling future breakthroughs in related open problems.

Conclusion

This work decisively demonstrates that the additive $O(\log\log N)$ term in Erdős's subset-sum threshold function $f(N)$ is unavoidable in general, disproving the conjecture that $f(N)\le \log_2 N+o(\log\log N)$. The proof leverages a sophisticated blend of probabilistic analysis, combinatorial enumeration, linear algebra, and AI-assisted synthesis to achieve a quantitatively precise lower bound for cyclic groups of prime order. This elucidates the inherent limitations in random subset sum coverage and sets the stage for deeper quantitative and algorithmic investigations in additive combinatorics and probabilistic group theory.