arXiv subjects

Mathias Løkkegaard Laursen

Publications and source records attributed to Mathias Løkkegaard Laursen.

4 recordsLinked to original sources

arXiv2025

Infinite products with algebraic numbers

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{α_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{α_{n,m}}\right)$, where the $α_n$ and $α_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities.

math.NT↗

arXiv2023

Attainable measures for certain types of p-adic Duffin--Schaeffer sets

This paper settles recent conjectures concerning the $p$-adic Haar measure applied to a family of sets defined in terms of Diophantine approximation. This is done by determining the spectrum of measure values for each family and seeing that this contradicts the corresponding conjectures.

math.NT↗

arXiv2022

Algebraic degree of series of reciprocal algebraic integers

In this paper, I give sufficient conditions for any linear combination in $\mathbb{Q}$ of numbers $\sum_{n=1}^{\infty}\frac{b_{1,n}}{α_{1,n}}$, $\ldots$, $\sum_{n=1}^{\infty}\frac{b_{K,n}}{α_{K,n}}$ to have algebraic degree greater than an arbitrary fixed integer $D$ when the numbers $α_{i,n}$ are algebraic integers of sufficiently rapidly increasing modulus and the $b_{i,n}$ are positive integers that are not too large.

math.NT↗

arXiv2022

The $p$-adic Duffin--Schaeffer conjecture

We prove Haynes' version of the Duffin--Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.

math.NT↗