We develop a recursive method for computing the $r$-removed $P$-orderings and $P$-orderings of order $h,$ the characteristic sequences associated to these and limits associated to these sequences for subsets $S$ of a Dedekind domain $D.$ This method is applied to compute these objects for $S=\mathbb{Z}$ and $S=\mathbb{Z}\setminus p\mathbb{Z}$.

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets $J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface $$𝒳\subset {\mathbb{P}}^{n+1}\times {\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}$$ parametrizing all projective hypersurfaces $X\subset {\mathbb{P}}^{n+1}\left(ℂ\right)$ of degree $d$. In 2004, for $k=n$, Siu announced that there exist two constants $c_n\ge 1$ and $c_n^{\prime }\ge 1$ such that the twisted tangent bundle $$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$ is generated at every point by its global sections. In...

Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes...

Let $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ and $g$ a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$. We write $f\to g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\to g$ for some $f:\left(\genfrac{}{}{0pt}{}{I}{m}\right)\to k$ is called the of $g$ and denoted by $w_k\left(g\right)$. In the first part of the paper we prove the existence of colorings with high $k$-width. In particular, we show that for each $k>0$ and $m>0$ there is a coloring $g$ with $w_k\left(g\right)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized...