arXiv subjects

Peter van Hintum

Publications and source records attributed to Peter van Hintum.

23 records · Page 2Linked to original sources

arXiv2020

Sharp Stability of Brunn-Minkowski for Homothetic Regions

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn-Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A \subset \mathbb{R}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A| \le d_n |A|$, then $|\operatorname{co}(A)\setminus A| \le C_n |\frac{A+A}{2}\setminus A|$ for $\operatorname{co}(A)$ the convex hull of $A$.

math.MG↗

arXiv2020

The Eternal Game Chromatic Number of Random Graphs

The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic number of random graphs. We show that with high probability $χ_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \mathbb{N}$. The upper bound applies for even $n$ and any other value of $p$ as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza.

math.CO↗

arXiv2019

(t,r) broadcast domination in the infinite grid

The $(t,r)$ broadcast domination number of a graph $G$, $γ_{t,r}(G)$, is a generalization of the domination number of a graph. $γ_{t,r}(G)$ is the minimal number of towers needed, placed on vertices of $G$, each transmitting a signal of strength $t$ which decays linearly, such that every vertex receives a total amount of at least $r$ signal. In this paper we prove a conjecture by Drews, Harris, and Randolph about the minimal density of towers in $\mathbb{Z}^2$ that provide a $(t,3)$ domination broadcast for $t>17$ and explore generalizations. Additionally, we determine the $(t,r)$ broadcast domination number of powers of paths, $P_n^{(k)}$ and powers of cycles, $C_n^{(k)}$.

math.CO↗

arXiv2019

Sharp quantitative stability of the planar Brunn-Minkowski inequality

We prove a sharp stability result for the Brunn-Minkowski inequality for $A,B\subset\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit $δ=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ is sufficiently small in terms of $t=|A|^{\frac{1}{2}}/(|A|^{\frac{1}{2}}+|B|^{\frac{1}{2}})$, there exist homothetic convex sets $K_A \supset A$ and $K_B\supset B$ such that $\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|} \le C t^{-\frac{1}{2}}δ^{\frac{1}{2}}$. The key ingredient is to show for every $ε>0$, if $δ$ is sufficiently small then $|co(A+B)\setminus (A+B)|\le (1+ε)(|co(A)\setminus A|+|co(B)\setminus B|)$.

math.FA↗

arXiv2018

Biased partitions of $\mathbb{Z}^n$

Given a function $f$ on the vertex set of some graph $G$, a scenery, let a simple random walk run over the graph and produce a sequence of values. Is it possible to, with high probability, reconstruct the scenery $f$ from this random sequence? To show this is impossible for some graphs, Gross and Grupel, call a function $f:V\to\{0,1\}$ on the vertex set of a graph $G=(V,E)$ $p$-biased if for each vertex $v$ the fraction of neighbours on which $f$ is 1 is exactly $p$. Clearly, two $p$-biased functions are indistinguishable based on their sceneries. Gross and Grupel construct $p$-biased functions on the hypercube $\{0,1\}^n$ and ask for what $p\in[0,1]$ there exist $p$-biased functions on $\mathbb{Z}^n$ and additionally how many there are. We fully answer this question by giving a complete characterization of these values of $p$. We show that $p$-biased functions exist for all $p=c/2n$ with $c\in\{0,\dots,2n\}$ and, in fact, there are uncountably many of them for every $c\in\{1,\dots,2n-1\}$. To this end, we construct uncountably many partitions of $\mathbb{Z}^n$ into $2n$ parts such that every element of $\mathbb{Z}^n$ has exactly one neighbour in each part. This additionally shows that not all sceneries on $\mathbb{Z}^n$ can be reconstructed from a sequence of values on attained on a simple random walk.

math.CO↗