Related papers: Goodman-Strauss theorem revisited
Let $G$ be a group with undecidable domino problem, such as $\mathbb{Z}^2$. We prove that all nontrivial dynamical properties for sofic $G$-subshifts are undecidable, that this is not true for $G$-SFTs, and an undecidability result for…
In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as…
Rice's theorem shows that nontrivial extensional properties of partial recursive functions are undecidable. For finite weighted Boolean optimization/CSP-style slices, a Rice-style structural analogue holds for tractability classification:…
Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $\omega$. Such sets…
Let $T$ be a theory with a definable topology. $T$ is t-minimal in the sense of Mathews if every definable set in one variable has finite boundary. If $T$ is t-minimal, we show that there is a good dimension theory for definable sets,…
Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch's problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition…
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The…
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The…
A generalization of topos theory is proposed giving an abstract realization of such categories as, say, the categories of manifolds and of Grothendieck schemes on the one hand, and permitting one, on the other hand, a view on…
We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt…
We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, F{\o}lner and dynamical…
We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably…
In the paper we introduce a weak set theory $\mathsf{H}_{<\omega}$ . A formalization of arithmetic on finite von Neumann ordinals gives an embedding of arithmetical language into this theory. We show that $\mathsf{H}_{<\omega}$ proves a…
Let $\Gamma$ be a sofic group, $\Sigma$ be a sofic approximation sequence of $\Gamma$ and $X$ be a $\Gamma$-subshift with nonnegative sofic topological entropy with respect to $\Sigma$. Further assume that $X$ is a shift of finite type, or…
A family of algebraic surfaces with many nondegenerate real singularities is introduced with the help of a construction, which has been used in previous works for the generation of substitution tilings.
A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit…
A fundamental result of K\"uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling.…
In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles…
We assumed that, for every natural number k, there is a natural number u such that the (k-1)th term of G(u) is k^k, and that G(u) terminates finitely. It immediately follows that every Goodstein Sequence G(m) over the natural numbers must…
We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\times m$ and ask whether or not a tiling of this region can be rearranged so…