Contributions to the theory of $F$-automatic sets

Fractal dimensions of $k$-automatic sets

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of "$k$-automatic" sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting…

Automata and tame expansions of $(\mathbb{Z},+)$

The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$ whose set of representations base $d$ is recognized by a finite automaton, a necessary condition is found for $x+y\in A$ to be a stable formula in $\operatorname{Th}(\mathbb{Z},+,A)$. Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the $d$-automatic $A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. This characterization is in terms of what were called "$F$-sets" by Moosa and Scanlon …

有理数的加法组没有自动表示

The additive group of the rationals does not have an automatic presentation

我们证明了有理数的加法群不具有自动表示。这个证明也适用于某些其他的阿贝尔群，例如，对于无穷多个素数$p$ -可分的无挠群$p$，或$Q/Z$。证明是组合和使用最显着的弗赖曼定理集与小倍。

F-sets and finite automata

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $Γ$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $Γ$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $Γ$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subva…

$k$-自动展开的Presburger算术的二元性

A Dichotomy for $k$-automatic expansions of Presburger Arithmetic

研究$ k $-自动集在Presburger算术扩展中的定义性，证明要么所有$ k $-自动集可定义，要么$(\mathbb{N},+,X)$与$(\mathbb{N},+,k^{\mathbb{N}})$定义集相同。

凝聚集与Solovay模型

Condensed Sets and the Solovay Model

本文通过几何形变和自动连续性，证明了在Solovay模型中离散凝聚阿贝尔群的Whitehead问题的解决，并展示了局部紧阿贝尔群之间的内部Ext计算。