Definability and decidability in expansions by generalized Cantor sets

TL;DR

This paper characterizes the definable sets in expansions of the ordered real additive group by generalized Cantor sets and shows that adding two such sets in multiplicatively independent bases leads to undecidability.

Contribution

It provides a complete description of definable sets in these expansions and proves the undecidability of the theory when two generalized Cantor sets are used in multiplicatively independent bases.

Findings

Definable sets in expansions by a single generalized Cantor set are characterized.

The theory remains decidable with one generalized Cantor set.

Adding two generalized Cantor sets in multiplicatively independent bases makes the theory undecidable.

Abstract

We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a set $C$ is a generalized Cantor set in base $r$ if there is a non-empty $K\subseteq\{1,\ldots,r-2\}$ such that $C$ is the set of those numbers in $[0,1]$ that admit a base $r$ expansion omitting the digits in $K$. While it is known that the theory of an expansion of the ordered real additive group by a single generalized Cantor set is decidable, we establish that the theory of an expansion by two generalized Cantor sets in multiplicatively independent bases is undecidable.