Achievement sets -- current results and open problems
We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.
math.CA↗
arXiv subjects
Szymon Głąb
Publications and source records attributed to Szymon Głąb.
Kenyon Theorem Revisited
We study sets $E(Σ,q)=\left\{\sum_{i=1}^\infty σ_iq^i\colon(σ_i)\inΣ^{\mathbb N}\right\}$ for a finite set $Σ\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|Σ|=1$ we prove several new equivalent conditions for $E(Σ,q)$ to contain an interval. We give a full characterization, if additionally $|Σ|$ is prime.
math.GN↗
On the structure of Borel ideals in-between the ideals $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$ in the Katětov order
For a family $\mathcal{F}\subseteq ω^ω$ we define the ideal $\mathcal{I}(\mathcal{F})$ on $ω\timesω$ to be the ideal generated by the family $\{A\subseteq ω\timesω:\exists f\in \mathcal{F}\,\forall^\infty n\, (|\{k:(n,k)\in A\}|\leq f(n))\}.$ Using ideals of the form $\mathcal{I}(\mathcal{F})$, we show that the structure of Borel ideals in-between two well known Borel ideals $\mathcal{ED} = \{A\subseteqω\timesω:\exists m \, \forall^\infty n\, (|\{k:(n,k)\in A\}|<m))\}$ and $\mathrm{Fin}\otimes\mathrm{Fin} = \{A\subseteqω\timesω:\forall^\infty n \, (|\{k:(n,k)\in A\}|<\aleph_0))\}$ in the Katětov order is fairly complicated. Namely, there is a copy of $\mathcal{P}(ω)/\mathrm{Fin}$ in-between $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$, and consequently there are increasing and decreasing chains of length $\mathfrak{b}$ and antichains of size $\mathfrak{c}$.
math.GN↗
Evolutions of finite graphs
Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We also study the corresponding profinite graphs. Finally, we present a concrete model of the projectively universal profinite graph, the projective Fraisse limit of finite graphs, showing in particular that it has a dense subset of isolated vertices.
math.CO↗
Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses
Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions $\{f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. We prove some results on Borel hull of Haar-$\mathcal I$ sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.
math.GN↗
Dense free subgroups of automorphism groups of homogeneous partially ordered sets
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense elements $\bar{g}\in\operatorname{Aut}(A)^m$ for the diagonal action is comeager in $\operatorname{Aut}(A)^m$. In our considerations we strongly use the result of Schmerl which says that there are essentially four types of countably infinite ultrahomogeneous posets.
math.GR↗
On the discrete boundary value problem for anisotropic equation
Using critical point theory methods we undertake the existence and multiplicity of solutions for discrete anisotropic two-point boundary value problems.
math.CA↗