On a different weighted zero-sum constant

TL;DR

The paper introduces the constant \(C_A(M)\), the least \(k\) such that every length-\(k\) sequence in a finite \(R\)-module \(M\) has an \(A\)-weighted zero-sum subsequence of consecutive terms. For \(M=\mathbb Z_n\), it determines \(C_A(n)\) for several weight sets \(A\), including \(U(n)\), \(U(n)^2\), and \(U(n)^3\) under stated conditions.

Key findings

• For any odd natural number \(n\), \(C_{U(n)}(n)=2^{\Omega(n)}\) (Corollary 4).
• For a prime \(p\), \(C_{U(p)^2}(p)=3\) when \(p\neq 2\), and \(C_{U(2)^2}(2)=2\); if every prime divisor of \(n\) is at least \(7\), then \(C_{U(n)^2}(n)=3^{\Omega(n)}\) (Theorem 4, Corollary 6).
• For \(p\equiv 1\pmod 3\), \(p\neq 7\), \(C_{U(p)^3}(p)=D_{U(p)^3}(p)=3\); also \(C_{U(7)^3}(7)=4\) and \(D_{U(7)^3}(7)=3\). For squarefree \(n\) not divisible by \(2,7,13\), \(C_{U(n)^3}(n)=2^{\Omega(n_2)}3^{\Omega(n_1)}\) (Theorem 7, Lemma 2, Corollary 8).
• General lower bound: if \(n=m_1m_2\), \(A\subseteq\mathbb Z_n\), \(A_1\subseteq\mathbb Z_{m_1}\), \(A_2\subseteq\mathbb Z_{m_2}\), and \(f_{n,m_1}(A)\subseteq A_1\), \(f_{n,m_2}(A)\subseteq A_2\), then \(C_A(n)\ge C_{A_1}(m_1)C_{A_2}(m_2)\) (Lemma 3). Also \(C_A(n)=2\) for \(A=\mathbb Z_n\setminus\{0\}\) (Theorem 2).

Methods

• Theoretical combinatorial number theory: defines \(A\)-weighted zero-sum sequences and the constants \(D_A(M)\) and \(C_A(M)\), then proves exact values and bounds for \(\mathbb Z_n\) with weight sets such as \(U(n)\), \(U(n)^2\), \(U(n)^3\), and \(\mathbb Z_n\setminus\{0\}\).
• No empirical data or sample; arguments use pigeonhole, Cauchy–Davenport, group/ring homomorphisms, Chinese remainder reductions, and cited lemmas on units and weighted zero-sums.
• Main analysis is case-based, often by induction on \(\Omega(n)\), reducing modulo prime powers and using observations relating weighted zero-sum sequences in \(\mathbb Z_n\) to their images in \(\mathbb Z_{p^{v_p(n)}}\).

Limitations

• The concluding remarks state that the even-\(n\) case for \(A=U(n)\) remains to be investigated, and whether the lower bounds in Corollaries 5 and 7 are exact for \(A=U(n)^2\) and \(A=U(n)^3\) is left open.
• Several results have stated exceptions: Lemma 1 fails for \(p=7,13\); Lemma 7 may fail for \(p<7\); Corollary 9 is false for \(p=2,3,7\); and Theorem 10/Corollary 8 are not true when \(n\) is divisible by \(7\).
• Evident from the text: many exact determinations are conditional on divisibility or prime-divisor restrictions, and no general exact formula is given for all \(n\) for \(U(n)^2\) or \(U(n)^3\).