arXiv subjects
Publications and source records attributed to Afshin Zarei.
For any irrational number $ α>\frac{3+\sqrt{5}}{2}\approx2.618$ and given a positive $ n\in\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $α$.
We introduce a recursive theory that completely axiomatizes the structure $\langle \mathbb{Z},<, +,f,0\rangle$ where $f$ is the function that maps each $x$ to the integer part of $φx $, with $φ$ the golden ratio. We prove that our axiomatization is model-complete in a language expanded with a function which we which we refer as the Fibonacci floor function.
We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$ to define it.
We introduce a model-complete theory which completely axiomatizes the structure $Z_α=(Z, +, 0, 1, f)$ where $f : x \to \lfloorα x \rfloor $ is a unary function with $α$ a fixed transcendental number. When $α$ is computable, our theory is recursively enumerable, and hence decidable as a result of completeness. Therefore, this result fits into the more general theme of adding traces of multiplication to integers without losing decidability.
We have provided a pure model-theoretic proof for the decidability of the additive structure of the integers together with the function {f} sending {x} to {[ϕx]} where ϕ is the golden ratio.