arXiv subjects

Peter Michael Reichstein Rasmussen

Publications and source records attributed to Peter Michael Reichstein Rasmussen.

4 recordsLinked to original sources

arXiv2026

Powers in prime bases and a problem on central binomial coefficients

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do so we examine numbers of the form $α^{k}$ in base $p$, where $p$ is a prime and $(α, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$'s less than $a$ such that $(α^{k})_p$ contains less than $n$ digits greater than $\frac{p}{2}$. This is done by showing that every sequence of the form $\langle σ_t, \dots, σ_1,σ_0 \rangle$, where $0\leq σ_i<p$ for $i\geq 1$ and $σ_0$ is in the residue class generated by $α$ modulo $p$, occurs at specific places in the representation $(α^k)_p$ as $k$ varies.

math.NT↗

arXiv2019

Classifying Convex Bodies by their Contact and Intersection Graphs

Suppose that $A$ is a convex body in the plane and that $A_1,\dots,A_n$ are translates of $A$. Such translates give rise to an intersection graph of $A$, $G=(V,E)$, with vertices $V=\{1,\dots,n\}$ and edges $E=\{uv\mid A_u\cap A_v\neq \emptyset\}$. The subgraph $G'=(V, E')$ satisfying that $E'\subset E$ is the set of edges $uv$ for which the interiors of $A_u$ and $A_v$ are disjoint is a unit distance graph of $A$. If furthermore $G'=G$, i.e., if the interiors of $A_u$ and $A_v$ are disjoint whenever $u\neq v$, then $G$ is a contact graph of $A$. In this paper we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies $A$ and $B$ are equivalent if there exists a linear transformation $B'$ of $B$ such that for any slope, the longest line segments with that slope contained in $A$ and $B'$, respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of $A$ and $B$ are the same if and only if $A$ and $B$ are equivalent. We prove the same statement for unit distance and intersection graphs.

cs.CG↗

arXiv2015

Flow Equivalence of Shift Spaces

We study two problems related to flow equivalence of shift spaces. The first problem, the classification of $S$-gap shifts up to flow equivalence, is partially solved with the establishment of a new invariant for the sofic $S$-gap shifts and a complete classification of the non-sofic $S$-gap shifts. The second problem is an examination of the entropy of shift spaces under flow equivalence. For a wide array of classes of shift spaces with non-zero entropy, it is shown that the entropies achievable while maintaining flow equivalence are dense in $\mathbb R^+$.

math.DS↗

arXiv2015

On a Conjecture on the Representation of Positive Integers as the Sum of Three Terms of the Sequence $\left\lfloor \frac{n^2}{a} \right\rfloor$

We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his original paper.

math.NT↗