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We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (n) . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define X (K , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]

Denote the set of algebraic numbers as ℚ ¯ and the set of algebraic integers as ℤ ¯. For γ ∈ ℚ ¯ , consider its irreducible polynomial in ℤ [ x ] , F γ (x) = a n x n + ⋯ + a 0 . Denote e (γ) = gcd (a n , a n − 1 , ... , a 1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ [ γ ] ∩ ℚ = { α ∈ ℚ | { p | v p (α) < 0 } ⊆ { p : p | e (γ) } }. Given a number field K and γ ∈ ℚ ¯ , we show that there is a subset X (K , γ) ⊆ Spec ( K) , for which K [ γ ] ∩ K = { α ∈ K | { | v (α) < 0 } ⊆ X (K , γ) }. We prove that K [ γ ] ∩ K is a principal ideal domain if and only if the primes in X (K , γ) generate the class group of K . We show that given γ ∈ ℚ ¯ , we can find a finite set S ⊆ ℤ ¯ , such that for every number field K , we have X (K , γ) = { ∈ Spec ( K) | p ∩ S ≠ ∅ }. We study how this set S relates to the ring ℤ ¯ [ γ ] and the ideal γ = { a ∈ ℤ ¯ | a γ ∈ ℤ ¯ } of ℤ ¯. We also show that γ 1 , γ 2 ∈ ℚ ¯ satisfy γ 1 = γ 2 if and only if X (K , γ 1) = X (K , γ 2) for all number fields K. [ABSTRACT FROM AUTHOR]

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to 3 f 4 . We denote the number of numerical semigroups of Frobenius number f by N(f). While N(f) is not monotonic we prove that N (f) < N (f + 2) for every f. [ABSTRACT FROM AUTHOR]

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F , genus g and type t. It is known that for any numerical semigroup g F + 1 − g ≤ t ≤ 2 g − F. Numerical semigroups with t = 2 g − F are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with t = g F + 1 − g . We show that for a fixed α the number of numerical semigroups with Frobenius number F and type F − α is eventually constant for large F. The number of numerical semigroups with genus g and type g − α is also eventually constant for large g. [ABSTRACT FROM AUTHOR]

Correction to: Semigroup Forum https://doi.org/10.1007/s00233-022-10282-6 This article was updated to correct an error in the received date made during the production process. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. [Extracted from the article]

A numerical set T is a subset of N 0 that contains 0 and has finite complement. The atom monoid of T is the set of x ∈ N 0 such that x + T ⊆ T. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup S and show that numerical sets with atom monoid S are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when S has small type. [ABSTRACT FROM AUTHOR]