Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Badly approximable numbers over imaginary quadratic fields

Abstract: We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the "badly approximable" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\geq C/|q|^2$ for all $p/q\in K$), by boundedness of the partial quotients in the continued fraction expansion of $z$. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in $\mathbb{C}$ whose points are badly approximable over $K$, with effective constants. By other methods (the Dani correspondence), we prove the existence of circles of badly approximable numbers over any imaginary quadratic field, with loss of effectivity. Among these badly approximable numbers are algebraic numbers of every even degree over $\mathbb{Q}$, which we characterize. All of the examples we consider are associated with cocompact Fuchsian subgroups of the Bianchi groups $SL_2(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers in an imaginary quadratic field.

Euler measure as generalized cardinality

Abstract: Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sketch some ideas that might be used in further development of a theory along lines suggested by Schanuel.

Diophantine quadruples in the ring of integers of the pure cubic field $Q(\sqrt[3]{2})$

Abstract: We show that in the ring of integers of the pure cubic field Q( 3 √ 2) there exits a D(w)-quadruple if and only if w can be represented as a difference of two squares of integers in Q( 3 √ 2).

Dynamical systems in unramified or totally ramified extensions of a \mathfrak{p}-adic field

Abstract: We use local field theory to study a special class of discrete dynamical systems, where the function being iterated is a polynomial whose coefficients belong to the ring of integers in a -adic field.

Codes over rings of size $p^2$ and lattices over imaginary quadratic fields

Abstract: Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $O_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $O_K / p O_K$ determine lattices $\Lambda_\ell (C) $ over $K$. If $ p mid \ell$ then the ring $\R:=O_K / p O_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, theta functions on the corresponding lattices are defined. These theta series $\theta_{\Lambda_{\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\ell \frac {p(n+1)(n+2)} 2$ there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes $p< 7$ and $\ell \leq 59$.