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Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows., Comment: 10 pages. Comments welcome

We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by $10$ and this bound is optimal, extending a result of Rampersad, who proved that the bound $9$ holds for the Fibonacci word. We then give a general result showing that for every $e \in [1,(5+\sqrt{5})/2)$ there is a natural number $N$, depending only on $e$, such that every Sturmian word has the property that the distance between consecutive ending positions of $e$-powers occurring in the word is uniformly bounded by $N$.

We study point modules of monomial algebras associated with symbolic dynamical systems, parametrized by proalgebraic varieties which 'linearize' the underlying dynamical systems. Faithful point modules correspond to transitive sub-systems, equivalently, to monomial algebras associated with infinite words. In particular, we prove that the space of point modules of every prime monomial algebra with Hilbert series $1/(1-t)^2$ -- which is thus thought of as a 'monomial $\mathbb{P}^1$' -- is isomorphic to a union of a classical projective line with a Cantor set. While there is a continuum of monomial $\mathbb{P}^1$'s with non-equivalent graded module categories, they all share isomorphic parametrizing spaces of point modules. In contrast, free algebras are geometrically rigid, and are characterized up to isomorphism from their spaces of point modules. Furthermore, we derive enumerative and ring-theoretic consequences from our analysis. In particular, we show that the formal power series counting the irreducible components of the moduli schemes of truncated point modules of finitely presented monomial algebras are rational functions, and classify isomorphisms and automorphisms of projectively simple monomial algebras.

Let $F$ be an algebraically closed field of positive characteristic and let $R$ be a finitely generated $F$-algebra with a filtration with the property that the associated graded ring of $R$ is an integral domain of Krull dimension two. We show that under these conditions $R$ satisfies a polynomial identity, answering a question of Etingof in the affirmative in a special case., Comment: 9 pages

This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a B\"uchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure $(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ for some $r \in \mathbb{N}_{>1}$. In the case that the closure of the predicate has Hausdorff dimension less than $1$, the dichotomy further characterizes these expansions of $(\mathbb{R},<,+,0,1)$ by when they have NIP and NTP$_2$, which is precisely when the closure of the predicate has Hausdorff dimension $0$., Comment: 25 pages

Let $\Bbbk$ be a field of characteristic zero and let $\mathbb{W}_n = \operatorname{Der}(\Bbbk[x_1,\cdots,x_n])$ be the $n^{\text{th}}$ general Cartan type Lie algebra. In this paper, we study Lie subalgebras $L$ of $\mathbb{W}_n$ of maximal Gelfand--Kirillov (GK) dimension, that is, with $\operatorname{GKdim}(L) = n$. For $n = 1$, we completely classify such $L$, proving a conjecture of the second author. As a corollary, we obtain a new proof that $\mathbb{W}_1$ satisfies the Dixmier conjecture, in other words, $\operatorname{End}(\mathbb{W}_1) \setminus \{0\} = \operatorname{Aut}(\mathbb{W}_1)$, a result first shown by Du. For arbitrary $n$, we show that if $L$ is a GK-dimension $n$ subalgebra of $\mathbb{W}_n$, then $U(L)$ is not (left or right) noetherian., Comment: 16 pages. Comments welcome!

Let $X$ be an algebraic variety equipped with a dominant rational self-map $\phi:X\to X$. A new quantity measuring the interaction of $(X,\phi)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,\phi)$ captures the maximum number of new algebraically independent invariant rational functions on the cartesian product of $(X, \phi)$ and $(Y, \psi)$, as $(Y,\psi)$ ranges over all algebraic dynamical systems. It is shown that this birational invariant agrees with the maximum dimension of a dominant equivariant rational image $(X',\phi')$ where $\phi'$ is part of an algebraic group action on $X'$. As a consequence, it is deduced that if some cartesian power of $(X,\phi)$ admits a nonconstant invariant rational function, then already the second cartesian power does., Comment: 20 pages, to appear in Selecta

We prove a result that can be seen as an analogue of the P\'olya-Carlson theorem for multivariate D-finite power series with coefficients in $\bar{\mathbb{Q}}$. In the special case that the coefficients are algebraic integers, our main result says that if $$F(x_1,\ldots ,x_m)=\sum f(n_1,\ldots ,n_m)x_1^{n_1}\cdots x_m^{n_m}$$ is a D-finite power series in $m$ variables with algebraic integer coefficients and if the logarithmic Weil height of $f(n_1,\ldots ,n_m)$ is $o(n_1+\cdots +n_m)$, then $F$ is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of $F$ has the form $1-\zeta x_1^{q_1}\cdots x_m^{q_m}$ where $\zeta$ is a root of unity and $q_1,\ldots ,q_m$ are nonnegative integers, not all of which are zero.

We associate lattices to the sets of unions and intersections of left and right quotients of a regular language. For both unions and intersections, we show that the lattices we produce using left and right quotients are dual to each other. We also give necessary and sufficient conditions for these lattices to have maximal possible complexity., Comment: In Proceedings AFL 2023, arXiv:2309.01126

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.

A theorem of Cobham says that if $k$ and $\ell$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $\ell$-automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse $k$-automatic subset of $\mathbb{N}^d$ and a sparse $\ell$-automatic subset of $\mathbb{N}^d$ is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets., Comment: 14 pages

We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \, : \, n=a+b+c \text{ and } a,b,c \in A \}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing., Comment: additional author added; largely rewritten with different example

We study quantum determinantal rings at roots of unity and calculate the PI degree using results of Lenagan-Rigal and Haynal to reduce the problem to finding properties of their associated matrices. These matrices, in turn, correspond to Cauchon-Le diagrams from which we can calculate the required matrix properties. In particular, we show that any matrix corresponding to an $m\times n$ diagram has invariant factors which are powers of 2. Our calculations allow us to state an explicit expression for the PI degree of quantum determinantal rings when the deformation parameter $q$ is a primitive $\ell^{\text{th}}$ root of unity with $\ell$ odd. Using this newly calculated PI degree we present a method to construct an irreducible representation of maximal dimension. Building on these results, we use the strong connection between quantum determinantal rings and certain quantum Schubert varieties through noncommutative dehomogenisation to obtain expressions for the PI degree of such quantum Schubert varieties under the same conditions on $q$., Comment: 36 pages

We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Bartholdi. Finally, we show that non-amenable monomial algebras must contain noncommutative free subalgebras. Examples are given to emphasize the sharpness and necessity of the assumptions in our results., Comment: Accepted to IMRN

A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if $X$ is a projective variety over an algebraically closed field of characteristic zero which admits a global algebraic vector field $v:X\to TX$ such that $(X,v)$ has no proper invariant subvarieties then $X$ is an abelian variety. Vector fields on abelian varieties with this property are also examined., Comment: 7 pages

We consider Ore extensions of the form $T:=R[x;\sigma,\delta]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$., Comment: 13 pages

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $\Phi$. Let $\alpha\in G(K)$ and let $\Gamma\subseteq G(K)$ be a finitely generated subgroup. We show that the set $\{n\in\mathbb{N}\colon \Phi^n(\alpha)\in \Gamma\}$ is a union of finitely many arithmetic progressions along with a set of Banach density equal to $0$. In addition, assuming $\Phi$ is regular, we prove that the set $S$ must be finite., Comment: 12 pages

The (left) linear hull of a weighted automaton over a field is a topological invariant. If the automaton is minimal, the linear hull can be used to determine whether or not the automaton is equivalent to a deterministic one. Furthermore, the linear hull can also be used to determine whether the minimal automaton is equivalent to an unambiguous one. We show how to compute the linear hull, and thus prove that it is decidable whether or not a given automaton over a number field is equivalent to a deterministic one. In this case we are also able to compute an equivalent deterministic automaton. We also show the analogous decidability and computability result for the unambiguous case. Our results resolve a problem posed in a 2006 survey by Lombardy and Sakarovitch., Comment: Completely restructured based on reviewer feedback

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $\Phi\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $\Phi^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $\tau: G\to G_0$ of semiabelian varieties such that $\tau\circ \Phi^{m k}=\tau$ for some $k \geqslant 1$. We show that under these assumptions $\Phi$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp., Comment: 15 pages

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

We prove a general criterion for an irrational power series $f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n$ with coefficients in a number field $K$ to admit the unit circle as a natural boundary. As an application, let $F$ be a finite field, let $d$ be a positive integer, let $A\in M_d(F[t])$ be a $d\times d$-matrix with entries in $F[t]$, and let $\zeta_A(z)$ be the Artin-Mazur zeta function associated to the multiplication-by-$A$ map on the compact abelian group $F((1/t))^d/F[t]^d$. We provide a complete characterization of when $\zeta_A(z)$ is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on $\mathbb{R}^d/\mathbb{Z}^d$ in which Baake, Lau, and Paskunas prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier together with certain patching arguments involving linear recurrence sequences.

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. We show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then {there is} a multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ \Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various applications of our results., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2005.04281

We consider D-finite power series $f(z)=\sum a_n z^n$ with coefficients in a number field $K$. We show that there is a dichotomy governing the behaviour of $h(a_n)$ as a function of $n$, where $h$ is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either $f(z)$ is rational or $h(a_n)>[K:\mathbb{Q}]^{-1}\cdot \log(n)+O(1)$ for $n$ in a set of positive upper density and this is best possible when $K=\mathbb{Q}$., Comment: Minor change in the proof of Proposition 4.1

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $\phi$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ \phi=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorphisms $\phi$ of dynamical degree $1$ of semiabelian varieties $X$. Also, we prove a related result for regular dominant self-maps $\phi$ of semiabelian varieties $X$: assuming $\phi$ does not preserve a non-constant rational function, we have that the dynamical degree of $\phi$ is larger than $1$ if and only if the union of all $\phi$-invariant proper subvarieties of $X$ is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties., Comment: 13 pages

Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to \mathbb{P}^1$ be a rational map defined over $L$, and let $c\in \mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\mathfrak{p}}$ for which there exists a positive integer $a=a(\mathfrak{p})$ with the property that for $i\in \{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in K_{\mathfrak{p}}[[t]]$ that converges on the closed unit disc of $K_{\mathfrak{p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large $n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps $(h,g)$ of $\mathbb{P}^1 \times X$ with $g$ \'etale., Comment: 12 pages

A multivariate, formal power series over a field $K$ is a B\'ezivin series if all of its coefficients can be expressed as a sum of at most $r$ elements from a finitely generated subgroup $G \le K^*$; it is a P\'olya series if one can take $r=1$. We give explicit structural descriptions of $D$-finite B\'ezivin series and $D$-finite P\'olya series over fields of characteristic $0$, thus extending classical results of P\'olya and B\'ezivin to the multivariate setting.

Given a finite alphabet $\Sigma$ and a right-infinite word $w$ over the alphabet $\Sigma$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work up to an equivalence in which two words are equivalent if they have the exact same set of factors (finite contiguous subwords). We show that ${\rm Rec}(w)$ can be endowed with a natural topology and we show that if $w$ is word of linear factor complexity then ${\rm Rec}(w)$ is a finite topological space. In addition, we note that there are examples which show that if $f:\mathbb{N}\to \mathbb{N}$ is a function that tends to infinity as $n\to \infty$ then there is a word whose factor complexity function is ${\rm O}(nf(n))$ such that ${\rm Rec}(w)$ is an infinite set. Finally, we pose a realization problem: which finite topological spaces can arise as ${\rm Rec}(w)$ for a word of linear factor complexity?, Comment: 14 pages

Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index free subgroup (using a result of Dunwoody). In this work we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then $\limsup_n a_n^{1/n}$ exists and is either $1$, $2$, or at least $2\sqrt{2}$., Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1805.08118

Given a right-infinite word $\bf x$ over a finite alphabet $A$, the rank of $\bf x$ is the size of the smallest set $S$ of words over $A$ such that $\bf x$ can be realized as an infinite concatenation of words in $S$. We show that the property of having rank two is decidable for the class of $k$-automatic words for each integer $k\ge 2$.

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation., Comment: To appear in Proc. Lond. Math. Soc

Let $X$ be a projective variety and let $f$ be a dominant endomorphism of $X$, both of which are defined over a number field $K$. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ eventually stabilizes, where $Y\subset X$ is a subvariety invariant under $f$. We show this question has an affirmative answer when the map $f$ is \'etale. We also look at a related problem of showing that there is some integer $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geq 0$, we necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. We prove this holds for \'etale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on $\mathbb{P}^1$ where we allow for composition by multiple different maps $f_1,\dots,f_r$., Comment: 27 pages

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field that do not have the same set of preperiodic points, then the semigroup generated by f and g contains a nonabelian free semigroup., Comment: 16 pages

Given a finite alphabet $\Sigma$ and a right-infinite word $\bf w$ over $\Sigma$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to \mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under cyclic shift) of length-$n$ factors $x$ of $\bf w$ with the property that every element of the conjugacy class appears in $\bf w$. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity, and as a result we show that words of linear factor complexity have at most finitely many primitive factors $y$ with the property that $y^n$ is again a factor for every $n$. We then look at automatic sequences and show that the Lie complexity function of a $k$-automatic sequence is again $k$-automatic., Comment: 13 pages

We consider applications of a finitary version of the Affine Representability theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne. Using this result we are able to show that when given a finite set of polynomial identities, there is an algorithm that terminates after a finite number of steps which decides whether these identities force a ring to be commutative. We then revisit old commutativity theorems of Jacobson and Herstein in light of this algorithm and obtain general results in this vein. In addition, we completely characterize the homogeneous multilinear identities that imply the commutativity of a ring., Comment: 30 pages, to appear in Israel J. Math. Proposition 5.2 added in this version; title changed from earlier version; Acknowledgment updated

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\cap\Gamma$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $\Gamma$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $\Gamma$. The approach is to use finite automata to give a concrete description of $X\cap \Gamma$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $\Gamma=G(K)$. As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in $X(K)$ of bounded height. As an application of the effective description of $X\cap\Gamma$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset?

We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings., Comment: 18 pages, survey paper

Let $q$ a prime power and ${\mathbb F}_q$ the finite field of $q$ elements. We study the analogues of Mahler's and Koksma's classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then there are a multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ \Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then obtain results about the coefficients of $D$-finite power series using these facts., Comment: 29 pages

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(\Phi^n(x))$, where $\Phi\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $\Phi$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(\Phi^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.

Given a natural number $k\ge 2$ and a $k$-automatic set $S$ of natural numbers, we show that the lower density and upper density of $S$ are recursively computable rational numbers and we provide an algorithm for computing these quantities. In addition, we show that for every natural number $k\ge 2$ and every pair of rational numbers $(\alpha,\beta)$ with $0<\alpha<\beta<1$ or with $(\alpha,\beta)\in \{(0,0),(1,1)\}$ there is a $k$-automatic subset of the natural numbers whose lower density and upper density are $\alpha$ and $\beta$ respectively, and we show that these are precisely the values that can occur as the lower and upper densities of an automatic set., Comment: 16 pages. This version corrects the proof of Lemma 3.1 in addition to making other changes

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(\theta^n)$ with $\theta

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings., Comment: 19 pages