Factorization in additive monoids of evaluation polynomial semirings
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Cited by 3 publications
References 27 publications
A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.
A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.
On the Set of Betti Elements of a Puiseux Monoid
Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.
Representations of a number in an arbitrary base with unbounded digits
In this paper, we prove that, for β ∈ ℂ {\beta\in{\mathbb{C}}} , every α ∈ ℂ {\alpha\in{\mathbb{C}}} has at most finitely many (possibly none at all) representations of the form α = d n β n + d n - 1 β n - 1 + … + d 0 {\alpha=d_{n}\beta^{n}+d_{n-1}\beta^{n-1}+\dots+d_{0}} with nonnegative integers n , d n , d n - 1 , … , d 0 {n,d_{n},d_{n-1},\dots,d_{0}} if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ {{\mathbb{Q}}} (possibly β itself) in the real interval ( 1 , ∞ ) {(1,\infty)} . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ ( 1 , ∞ ) {{\mathbb{C}}\setminus(1,\infty)} , there is α ∈ ℚ ( β ) {\alpha\in{\mathbb{Q}}(\beta)} with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka.
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