### A characterization of minimal zero-sequences of index one in finite cyclic groups.

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Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

It is proved that if a finite abelian group is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be periodic. This result generalizes Hajós's factorization theorem.

Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. We prove, among other results, that $\u2081\left({C}_{n\u2081}\oplus {C}_{n\u2082}\right)=n\u2081+n\u2082$ for all integers n₁,n₂ with 1 < n₁|n₂.

Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. In this article, it is proved that for every prime p, $\u2081\left({C}_{p}\oplus {C}_{p}\right)=2p$, and it is also proved that $\u2081\left({C}_{mp}\oplus {C}_{mp}\right)=2mp$ if $\u2081\left({C}_{m}\oplus {C}_{m}\right)=2m$ and m is large enough. In particular, it is shown that for...

Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$. In other words, there exists a non-trivial character $\chi \in \widehat{G}$ such that ${\sum}_{a\in A}\chi \left(a\right)\in P$.

A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with ${Z}_{n}=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):{a}^{k}\equiv b\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral ⇒ tile" of the Spectral Set Conjecture, for all sets A of size |A| ≤ 5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd.