Two local zero-sum problems

Abstract: In the present paper, we investigate two local zero-sum problems. Let $n,k\ge 2$. We denote by $\mathsf{D}<sup>*(n,nk)$ (resp. $η<sup>{*}(n,nk)$) the smallest positive integer $\ell$ (if exists) such that, from any given $\ell$ integers not divisible by $n$, one can select some (resp. at most $n$) of them whose sum is divisible by $n$ but not by $nk$. We prove that both $\mathsf{D}<sup>*(n,nk)$ and $η<sup>{*}(n,nk)$ are equal to $2n-1$ if $\mathrm{rad}(n) \mid \mathrm{rad}(k)$ and infinite otherwise. The corresponding inverse problem is also determined. We denote by $\mathsf{D}_n<sup>{\times}$ (resp. $η_n<sup>{\times}$) the smallest positive integer $\ell$ such that, from any given $\ell$ integers coprime to $n$, one can select some (resp. at most $n$) of them whose sum $σ$ satisfies $\gcd(σ, n<sup>2)=n$. We prove that $\mathsf{D}_n<sup>{\times}=η_n<sup>{\times}=2n-1$ if $n$ is a prime power, and determine its inverse problem.

• The paper establishes that both D*(n,nk) and η*(n,nk) equal 2n-1 when rad(n) divides rad(k), while they become infinite otherwise.
• The methodology employs recursive decomposition, canonical homomorphism reductions, and explicit constructions to analyze extremal sequences.
• The inverse problem is solved by characterizing sequences of length 2n-2 via congruence relations, deepening the understanding of additive combinatorial structures.

Authoritative Summary of "Two local zero-sum problems" (2607.11313)

Context and Motivation

This paper addresses two new local variants of classical zero-sum problems in the context of finite abelian groups, with a focus on cyclic groups of composite order. The zero-sum theory has deep connections to combinatorial number theory, additive combinatorics, and the structural study of extremal sequences, and is rooted in results such as the Erdős-Ginzburg-Ziv theorem and the Davenport constant. The classical approach ensures that long enough sequences contain prescribed zero-sum subsequences, with corresponding inverse problems characterizing the extremal sequences for which such subsequences are absent. Recent developments, particularly those concerning zero-sum subsequences with refined divisibility constraints, motivate the present investigations.

Definitions and Problem Statements

For integers $n, k\geq 2$, let $\mathsf{D}^*(n,nk)$ (resp. $\eta^*(n,nk)$) denote the minimum integer $\ell$ such that any $\ell$ integers not divisible by $n$ contain a subsequence (resp. of length at most $n$) whose sum is divisible by $n$ but not by $nk$. If no such $\ell$ exists, the value is set to $\mathsf{D}^*(n,nk)$0.

Furthermore, for integers coprime to $\mathsf{D}^*(n,nk)$1, $\mathsf{D}^*(n,nk)$2 (resp. $\mathsf{D}^*(n,nk)$3) denotes the minimum length $\mathsf{D}^*(n,nk)$4 such that any $\mathsf{D}^*(n,nk)$5 integers coprime to $\mathsf{D}^*(n,nk)$6 contain a subsequence (resp. of length at most $\mathsf{D}^*(n,nk)$7) whose sum $\mathsf{D}^*(n,nk)$8 satisfies $\mathsf{D}^*(n,nk)$9.

These definitions constitute local constraints refining the classical zero-sum conditions via divisibility by $\eta^*(n,nk)$0 and non-divisibility by $\eta^*(n,nk)$1, or precise $\eta^*(n,nk)$2-adic valuation.

Principal Results

The central results of the paper are as follows:

• Both $\eta^*(n,nk)$3 and $\eta^*(n,nk)$4 equal $\eta^*(n,nk)$5 if $\eta^*(n,nk)$6, and are infinite otherwise.
• Both $\eta^*(n,nk)$7 and $\eta^*(n,nk)$8 equal $\eta^*(n,nk)$9 if $\ell$0 is a prime power.
• The precise structure of extremal sequences for the associated inverse problems is determined: For length $\ell$1, the absence of the prescribed zero-sum subsequence is characterized explicitly in terms of congruence classes and multiplicities, reflecting highly structured sequence construction.

These results extend previous work (e.g., Sun (Sun, 16 Jun 2026)) on specific cases (such as $\ell$2) and generalize bounds related to local zero-sum constants.

Methodological Innovations

The proofs leverage:

• Refined group-theoretic and additive combinatorial arguments, including canonical homomorphism reduction to cyclic groups of smaller order.
• Recursive decomposition techniques, allowing for inductive proofs by considering subgroups and quotient groups.
• Structural dichotomy lemmas and combinatorial constructions, particularly for the prime power case, yielding exact values.
• Explicit sequence construction for extremal cases, supporting the inverse characterization.
• Utilization of classical results (e.g., the Erdős-Ginzburg-Ziv theorem, classical bounds for Davenport-type constants) alongside recent generalizations.

Notably, the equivalence between $\ell$3 and $\ell$4, and between $\ell$5 and $\ell$6 in the specified cases, is established, simplifying the evaluation of these invariants.

Numerical Values and Strong Claims

• The paper provides strong claims: For any $\ell$7, $\ell$8 and $\ell$9 are exactly $\ell$0 when $\ell$1 (including the classical $\ell$2 case).
• For prime powers, $\ell$3.
• If $\ell$4, these invariants are infinite, indicating a sharp structural dichotomy.

Inverse Problems and Extremal Structure

The inverse characterization establishes that, for sequences of length $\ell$5 (not divisible by $\ell$6 or coprime to $\ell$7, respectively), the absence of the specified zero-sum subsequence is equivalent to the sequence being partitioned into two parts congruent to $\ell$8 and $\ell$9 modulo $n$0, for suitable $n$1 with $n$2 coprime to $n$3. For the prime power case, corresponding congruence relations modulo $n$4 for the terms are derived. These explicit constructions affirm the extremal nature and the necessity of the $n$5 threshold.

Implications and Future Directions

Practically, these local zero-sum results advance the theory by concretely quantifying the length threshold for sequences guaranteeing refined divisibility properties—relevant for the study of invariant factors, combinatorial designs, and their applications in algebraic combinatorics and cryptography. Theoretically, they clarify the interaction between the arithmetic of $n$6 and $n$7, and suggest further avenues:

• Determination of $n$8 and $n$9 for composite $n$0 with $n$1, as addressed in the concluding example and open problem.
• Further refinement of zero-sum constants involving higher $n$2-adic valuations or more complex divisibility constraints.
• Potential generalization to non-cyclic and higher-rank abelian groups.

Future research may explore achieving exact values for composite $n$3, possibly through advanced decomposition methods or cross-number invariants.

Conclusion

This paper rigorously establishes precise values and structural characterizations for two local zero-sum invariants in cyclic groups, showing that the threshold $n$4 remains optimal under refined divisibility constraints for relevant cases. The explicit construction of extremal sequences answers the corresponding inverse problems completely, providing foundational results for further exploration in zero-sum theory and additive combinatorics.