
The main part of this talk we published in Studies in Weak Arithmetics, V. 3 in 2016. The union of the set of all spectra \((\lfloor n\alpha \rfloor)_{n\in\mathbb{N}^{\geq 1}}\), for \(\alpha\in\mathbb{R}^{\geq 0}\), and that of rational upper spectra minus one \((\lceil n\alpha\rceil 1)_{n\in\mathbb{N}^{\geq 1}}\), for \(\alpha\in\mathbb{Q}^{> 0}\), is characterized in \(\mathbb{N}^{(\mathbb{N}\setminus\{0\})}\) by additive, equivalently multiplicative nearlinearity. There are counterparts for finite initial segments of a sequence to be extendible to some inhomogeneous spectrum \((\lfloor n\alpha +\gamma\rfloor)_{n\in\mathbb{N}^{\geq 1}}\). The equivalences of both pairs of criteria are limitbased in one direction, and rely on induction in the other. We turn to weak arithmetic and compare these properties for functions over models \(M\) of Open Induction in the language \({\cal L}=\{+,\cdot,<,0,1\}\). We establish that homogeneous MNL continues to characterize the union of the two types of functions in these models, but the multiplicative formulation is now stronger than the additive in both homogeneous and inhomogeneous cases.
[This is due to Shepherdson's model $M_0$ having $\mathbb{Z}$ as a direct summand of its additive group.]
Next we prove that for any \(M\models \mbox{IOpen}\), the spectrum of \(\varphi=\frac{1+\sqrt{5}}{2}\in\mbox{RC}(M)\) has jumps \(2\) precisely on the range of the upper spectrum minus one of \(\varphi\), regardless of its (ir)rationality. We obtain some independence results using rationality of all real algebraic numbers in \(M_0\). Here the range of the golden Beatty spectrum is opendefinable in \({\cal L}_{\lfloor \frac{x}{y}\rfloor}\). The same holds for \(\lfloor\frac{a}{b}\lfloor\frac{c}{d}n\rfloor\rfloor\) in any Euclidean Division Ring. In the standard setting, if
\(\alpha>0,0<\beta<1\) and \(\alpha\beta\) is irrational, then \(\lfloor \lfloor n\alpha\rfloor\beta\rfloor=\lfloor n\alpha\beta\rfloor\) on a set of positive density. If in addition the standard \(\alpha\) and \(\beta\) are algebraic, this also holds for any \(\mathbb{Z}\)chain of \(M_0\). Examples of further variants in \(M\) and \(M_0\) are brought up.
Here are some further facts we will discuss. Let $F$ be an ordered field, $G$ a maximal discrete additive subgroup, and $D$ a maximal discrete subring of $F$. We point out contrasts like the following: although $F$ cannot have any elements of infinite distance to $G$, there are examples where, on the contrary, $F$ realizes some gaps of $D$. Shifting to definable Dedekind completeness for ordered fields, it is known to be successively weakened if we just required nonexistence of definable regular gaps and then disallowing parameters. Reducing in the opposite order, at least one side is sharp: there are 0definably complete ordered fields which are not real closed.

