Let $G$ be a finite abeilian group. A sequence $S$ with terms from $G$ is zero-sum if the sum of terms in $S$ equals zero. It is a minimal zero-sum sequence if no proper, nontrivial subsequence is zero-sum. The maximal length of a minimal zero-sum subsequence in $G$ is the Davenport constant, denoted $D(G)$. For a rank 2 group $G=C_n \oplus C_n$, it is known that $D(G)=2n-1$. However, the structure of all maximal length minimal zero-sum sequences remains open. If every such sequence contains a term with multiplicity $n-1$, then $C_n \oplus C_n$ is said to have Property B, and it is conjectured that this is true for all rank 2 groups $C_n \oplus C_n$.
In this paper, we show that Property B is multiplicative, namely, if $G=C_n \oplus C_n$ and $G=C_m \oplus C_m$ both satisfy Property B, with $m, n\geq 3$ odd and $mn>9$, then $C_{mn}\oplus C_{mn}$ satisfies Property B also. Combined with previous work in the literature, this reduces the question of establishing Property B to the prime cases, and in such case the complete structural description of the sequence follows.
Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties
Mixed sums of squares and triangular numbers (III)
本文证实了孙文的一个猜想：每个正整数是一个平方、一个奇平方和一个三角数的和。对任意正整数m，我们证明了p= 2 m +1是3模4的素同余当且仅当T_m=m（m+1）/2不能表示为两个奇平方和一个三角数之和，即，p^2=x^2+8（y^2+z^2），其中x，y，z均为非奇数。我们还证明了一个正整数不能写成一个奇数平方和两个三角数之和当且仅当它是2T_m（m>0）的形式，其中2 m +1没有素除子与3模4全等。
Let $C_n$ be the cyclic group of order $n$. In this paper, we provide the exact values of some zero-sum constants over $C_n \rtimes_s C_2$ where $s \not\equiv \pm1 \pmod n$, namely $η$-constant, Gao constant, and Erdős-Ginzburg-Ziv constant (the latter for all but a "small" family of cases). As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for groups of this form. We also solve the associated inverse problems by characterizing the structure of product-one free sequences over $C_n \rtimes_s C_2$ of maximum length.
The Davenport constant for a finite abelian group $G$ is the minimal length $\ell$ such that any sequence of $\ell$ terms from $G$ must contain a nontrivial zero-sum sequence. For the group $G=(\mathbb Z/n\mathbb Z)^2$, its value is $2n-1$, which is a classical result of Olson. The associated inverse question is to characterize those sequences of maximal length $2n-2$ that do not have a nontrivial zero-sum subsequence. A simple argument shows this to be equivalent to characterizing all minimal zero-sum sequences of maximal length $2n-1$, with a minimal zero-sum sequence being one that cannot h…
The inverse problem associated to the Erdős-Ginzburg-Ziv constant and the $η$-constant is solved for finite abelian groups of the form $C_2 \oplus C_2 \oplus C_{2n}$ where $n \ge 2$ is an integer.
Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c > 0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A| < (0.25 - c)|\mathbb{F}| + o(|\mathbb{F}|)$ as $|\mathbb{F}| \rightarrow \infty$. We contrast this with the result that such sets exist with size at least $0.25|\mathbb{F}| - o(|\mathbb{F}|)$ when $\mathbb{F}$ has characteristic $2$.
Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0},λ_{1}, \ldots, λ_{k-1}) \in \{-1,0,1\}^{k},~Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the restricted signed sumset of $A$. The direct problem for the restricted signed sumset $h\hat{}_{\underline{+}}A$ is to find the minimum number of elements in $h\hat{}_{\underline{+}}A$ in terms of $|A|$. The inverse problem for $h\hat{}_{\underline{+}}A$ is to determine the struct…
Let $Φ_n(X)$ and $Ψ_n(X)=\frac{X^{n}-1}{Φ_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$ such that $ 0 \leq l_{1} \leq φ(d_{1})-1 $ and $ 0 \leq l_{2} \leq φ(d_{2})-1 $, we show that \[\left \langle X^{l_{1}} Ψ_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} Ψ_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $ \langle \cdot, \cdot \rangle $ is the inner product on $\mathbb{Q}[X]$ defined by $ \langle \sum_{k} a_{k}X^{k},\sum_{k} b_{k}X…
