Related papers: Carrots for dessert

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and…

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched…

The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic…

This note presents a short, transparent proof of the theorem that every Euclidean quadratic form over a normed integral domain is an Aubry-Davenport-Cassels form. The theorem, as formulated in the note, allows besides quadratic terms also…

Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to…

Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On…

For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings…

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known…

In this paper we study rational functions of the form $ R_{n,a,c}(z) = z^n + \dfrac{a}{z^n} + c, $ with $n$ fixed and at least $3$, and hold either $a$ or $c$ fixed while the other varies. We locate some homeomorphic copies of the…

A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where…

We study discrete orderings in the real spectrum of a commutative ring by defining discrete prime cones and give an algebro-geometric meaning to some kind of diophantine problems over discretely ordered rings. Also for a discretely ordered…

M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve…

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to…

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the…

Here are two of our main results: Theorem 1. Let X be a normal space with dim X=n and m\geq n+1. Then the space C*(X,R^m) of all bounded maps from X into R^m equipped with the uniform convergence topology contains a dense G_{\delta}-subset…

This article focuses on the dynamics of the different tridimensional principal slices of the multicomplex Multibrot sets. First, we define an equivalence relation between those slices. Then, we characterize them in order to establish…

Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $\mu_1,\mu_2, \dots, \mu_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $\omega_i=\mu_i(\mathbb{R}^d)$ for $i\in…

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof…

In important papers on cake-cutting -- one of the key areas in fair division and resource allocation -- the measure-theoretical fundamentals are not fully correctly given. It is not clear (i) which family of sets should be taken for the…