We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$ , where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$ . Finally, we discuss asymptotic bounds towards generic points on the unit circle.

This paper develops a generalization about agreement in German copula constructions described in Coon et al. (2017), and proposes an analysis that ties it to other well-established hierarchy phenomena. Specifically, we show that “assumed-identity” copula constructions in German exibit both person and number hierarchy effects, and that these extend beyond the “non-canonical” or “inverse” agreement patterns described in previous work on copula constructions (e.g., Béjar and Kahnemuyipour 2017 and works cited there). We present experimental evidence to support this generalization, and then develop an account that unifies it with hierarchy phenomena in other languages, with a focus on PCC effects. Specifically, we propose that what German copula constructions have in common with PCC environments is that there are multiple accessible DPs in the domain of a single agreement probe, the lower of which is more featurally specified than the higher (see, e.g., Béjar and Rezac 2003, 2009; Anagnostopoulou 2005; Nevins 2007). We also offer an explanation as to why number effects are present in German copula constructions but notably absent in PCC effects. We then place our account within the broader context of constraints on predication structures.

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if $F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then $F(z)$ is either rational or it is transcendental over the field of meromorphic functions.

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$

$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$

$\unicode[STIX]{x1D6F9}(n)\rightarrow 0$

$n\rightarrow \infty$

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ be a prime and $b$ a primitive root of $p^2$. In this paper, we give an explicit formula for the number of times a value in $\{0,1,\ldots,b-1\}$ occurs in the periodic part of the base-$b$ expansion of $1/p^m$. As a consequence of this result, we prove two recent conjectures of Aragón Artacho et al. [‘Walking on real numbers’, Math. Intelligencer35(1) (2013), 42–60] concerning the base-$b$ expansion of Stoneham numbers.

We prove a result concerning power series $f\left( Z \right)\,\in \,\mathbb{C}\left[\!\left[ Z \right]\!\right]$ satisfying a functional equation of the form?

1

$$f\left( {{Z}^{d}} \right)\,=\,\sum\limits_{k=1}^{n}{\frac{{{A}_{k}}\left( Z \right)}{{{B}_{k}}\left( Z \right)}\,f{{\left( Z \right)}^{k}}},$$

where ${{A}_{k}}\left( Z \right),\,{{B}_{k}}\left( Z \right)\,\in \,\mathbb{C}\left[ Z \right]$ . In particular, we show that if $f\left( Z \right)$ satisfies a minimal functional equation of the above form with $n\,\ge \,2$ , then $f\left( Z \right)$ is necessarily transcendental. Towards a more complete classification, the case $n=\,1$ is also considered.

In this paper, we prove that a non–zero power series $F(z\text{)}\in \mathbb{C}\text{ }[[z]]$ satisfying

$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$

where $d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$ , with $A(z)\,\ne \,0$ and $\deg \,A(z),\,\deg \,B(z)\,<\,d$ is transcendental over $\mathbb{C}(z)$ . Using this result and a theorem of Mahler’s, we extend results of Golomb and Schwarz on transcendental values of certain power series. In particular, we prove that for all $k\,\ge \,2$ the series ${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$ is transcendental for all algebraic numbers $z$ with $\left| z \right|\,<\,1$ . We give a similar result for ${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$ . These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.

Let Ω(n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j = 0,1,…,m – 1, we have

with α = 1. Building on work of Kubota and Yoshida, we show that for m > 2 and any j = 0,1,…,m – 1, the error term is not o(xα ) for any α < 1.

The controlled atomic layer growth of SiO2 insulating layers on silicon surfaces might be achieved through the sequential reaction of SiCl4 and H2O:

(A) Si-Cl + H2O → Si-OH + HCl

(B) Si-OH + SiCl4 → Si-O-SiCl3 + HCl.

To explore this ABAB… binary reaction scheme, laser-induced thermal desorption, temperature-programmed desorption, and Auger electron spectroscopy techniques were utilized to measure the kinetics of H2O oxidation of a Si(111)7×7 surface that had been previously subjected to a saturation SiCl4 exposure. Reaction kinetics studies for the oxidation of the chlorinated surface revealed that the rate of oxygen gain and the rate of chlorine loss were equal at reaction temperatures between 200 K and 700 K. These results were consistent with a direct substitution reaction according to:

(A) Si-Cl + H2O → Si-OH + HCl.

Above 700 K, the amount of oxygen gain became progressively greater than the amount of chlorine loss. This behavior was associated with the thermal desorption of H2 and the resultant formation of new dangling bond sites for H2O adsorption. For all temperatures, the oxidation kinetics of the chlorinated surface were nearly equivalent to the kinetics for the oxidation of clean silicon. This surprising result indicates that chlorine sites and free dangling bond sites react with equal probability to H2O. The kinetics of SiCl4 deposition were also measured on a Si(111)7×7 surface previously exposed to a saturation H2O dose. This chlorination reaction occurred at a much slower rate and was not as amenable to UHV studies.