Related papers: Fuglede's conjecture for a union of two intervals
Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus…
We obtain new partial results supporting the spectral set conjecture in dimension 1.
The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].
Several results about the union-closed sets conjecture are presented.
Let $R$ be a commutative ring. Roughly speaking, we prove that an $R$-module $M$ is flat iff it is a direct limit of $R$-module affine algebraic varieties, and $M$ is a flat Mittag-Leffler module iff it is the union of its $R$-submodule…
Let $p$ be a prime number, it is shown that tiling and spectral sets coincide in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$ by considering equivalently symplectic spectral pairs. The main approach is still to analyze the zero set of the Fourier…
In this paper we introduce a superclass of split digraphs, which we call spine digraphs. Those are the digraphs D whose vertex set can be partitioned into two sets X and Y such that the subdigraph induced by X is traceable and Y is a stable…
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…
In the present paper, we prove that two discrete rectangular tori are isospectral if and only if they are isomorphic.
The HRT (Heil-Ramanathan-Topiwala) conjecture stipulates that the set of any finitely many time-frequency shifts of a non-zero square Lebesgue integrable function is linearly independent. The present work settles two special cases of this…
We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.
In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$.…
We provide conditions under which the union of two first-order theories has the amalgamation property.
In this paper, we prove that it is undecidable whether a set of two polycubes can tile $\mathbb{Z}^3$ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By…
A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t\colon t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$…
We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set.
Extending the methods of Metrebian (2018), we prove that any symmetric punctured interval tiles $\mathbb Z^3$. This solves two questions of Metrebian and completely resolves a question of Gruslys, Leader and Tan. We also pose a question…
We propose a strengthening of the conclusion in Tur\'an's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Tur\'an theorem by weakening its…
A set S is independent if no two vertices from S are adjacent. In this paper we prove that if F is a collection of maximum independent sets of a graph, then there is a matching from S-{intersection of all members of F} into {union of all…
What spectral conditions imply a graph contains a chorded cycle? This question was asked by R.J. Gould in 2022. We answer two modified versions of Gould's question by giving tight spectral conditions that imply the existence of doubly…