Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.0799

Even universal binary Hermitian lattices over imaginary quadratic fields

Byeong Moon Kim·Ji Young Kim·Poo-Sung Park
View original source ↗Read source document ↗Export metadata ↓

Abstract

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\Q{-m}$ which admits even universal binary Hermitian lattices.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Byeong Moon Kim, Ji Young Kim, Poo-Sung Park. 2009-02-19. Even universal binary Hermitian lattices over imaginary quadratic fields. https://doi.org/10.1515/form.2011.043

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Small improvements to the Ball-Rivoal theorem and its $p$-adic variant

We prove that the dimension of the $\mathbb{Q}$-linear span of the Riemann zeta values $ζ(3),ζ(5),\ldots,ζ(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s \geqslant s_0$. This result slightly refines the theorem of Rivoal (2000) or Ball--Rivoal (2001). The proof incorporates the arithmetic factor $Φ_n$ introduced by Zudilin (2001) into a slight generalization of the Ball--Rivoal hypergeometric construction. The asymptotic impact of the arithmetic factor $Φ_n$ was not anticipated prior to this work. Although this result is subsumed by a much stronger recent development by Fischler (2026), our approach yields a new result in the $p$-adic setting, slightly refining a theorem of Sprang (2020). We prove that the dimension of the $\mathbb{Q}$-linear span of the $p$-adic zeta values $ζ_p(3),ζ_p(5),\ldots,ζ_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any prime $p$ and any sufficiently large even integer $s \geqslant s_0(p)$.

math.NT↗
arXiv2026

A Fibonacci theorem for Collatz trajectories via modular graph structure

Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers $n$ in $\{1,\ldots,2^m\}$ have the property that none of the iterates $T(n),T^2(n),\ldots,T^{m-1}(n)$ lies in the residue class $4\pmod6$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(φ/2)^m$, $φ=(1+\sqrt{5})/2$. Equivalently, these are the odd $n\le2^m$ for which no two consecutive terms of $n,T(n),\ldots,T^{m-1}(n)$ are even. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\{1,2,4,5\}]$. Removing vertex $4$ from $G'$ yields a subgraph of spectral radius $φ$, against $ρ(G')=2$; the Fibonacci count follows from this spectral gap. We construct an explicit bijection $Ψ_m:\{1,\ldots,6\cdot2^m\}\to\mathcal{P}_m(G)$ onto the directed paths of length $m$ in $G$. We further show that no vertex of $G'$ is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy $1<\sqrt{2}<φ<2$. In particular, every positive cycle of $T$ must visit residue class $2\pmod6$, and a flow conservation identity forces this class to account for more than $18\%$ of the steps in any such cycle.

math.NT↗