6 problems

Let $(a,b)$ be coprime, let $\mathcal{U}_{a,b}$ denote the expansion of $(\mathbb{N},+,0,1)$ by the associated Ulam predicate, and let $\operatorname{Th}(\mathcal{U}_{a,b})$ be its…

Let $(a,b)$ be coprime, and let $\mathcal{U}_{a,b}$ be the expansion of $(\mathbb{N},+,0,1)$ by the predicate for the associated Ulam sequence. Ulam dichotomy. For every coprime…

Undecidability conjecture. The first-order theory of $(\mathbb{Z};<,+,g)$ is undecidable.

Let $\alpha$ be a non-quadratic scalar. Undecidability conjecture. $\alpha$-Presburger arithmetic sentences with three alternating blocks of quantifiers are undecidable. The paper…

Let $R=\Z[t]$ and $X=\N$. Define … where each $g_{\alpha(t)}$ is interpreted by … with $g_{\alpha(t)}(x)=0$ when $\alpha(t)=0$. Quantifier-elimination conjecture. Every…

Let $S_t$ be a parametric Presburger family, meaning a family of subsets of $\mathbb{N}^d$ definable over the natural numbers using quantifiers, boolean operations, and inequalitie…