Maiken Gravgaard, Mathias Løkkegaard Laursen, Victor Shirandami
(Aarhus University, Aarhus University, Manchester University)

Symposium

**Speaker:** Mathias Løkkegaard Laursen

**Title:** Attainable measures of $p$-adic Duffin--Schaeffer sets

**Abstract:** Let $\psi:\mathbb N\to[0,\infty)$. This talk is based on a resent paper, in which I determine the possible measure values of the sets \begin{equation*} \mathcal{B}^p(\psi) = \bigg\{\alpha\in\mathbb Z_p : |\alpha-x|_p \le \frac{\psi(h_1(x))}{h_1(x)} \text{ for }\infty\text{ many } x\in\mathbb Q\bigg\} \end{equation*} and \begin{equation*} \mathfrak{A}^p(\psi) = \bigg\{\alpha\in\mathbb Z_p : |\alpha-x|_p \le \frac{\psi(h_2(x))}{h_2(x)} \text{ for }\infty\text{ many } x\in\mathbb Q\bigg\} \end{equation*} with respect to the $p$-adic Haar measure, where $h_1(a/b)=\max\{|a|,|b|\}$ and $h_2(a,b) = |ab|$ when $a$ and $b$ are coprime integers. This contradicts conjectures presented by Kristensen and myself in 2023 and by Badziahin and Bugeaud in 2022, respectively. Both sets $\mathcal{B}^p(\psi)$ and $\mathfrak{A}^p(\psi)$ are inspired by the set \begin{align*} \mathcal{A}(\psi) &= \bigg\{\alpha\in[0,1] : \bigg|\alpha-\frac{a}{n}\bigg| \le \frac{\psi(n)}{n} \text{ for }\infty\text{ many coprime } (a,n)\in\mathbb Z\times\mathbb N\bigg\}, \end{align*} which is known from the famous Duffin--Schaeffer Theorem (formerly known as the Duffin--Schaeffer Conjecture).

**Speaker:** Victor Shirandami

**Title:** Distribution of Algebraic projective points: Towards Probablistic Effectivity in The Subspace Theorem.

**Abstract:** In recent work of Koleda, an explicit formula for the limiting distribution of the real algebraic numbers of bounded naive height is obtained, with sharp error term. We shall first discuss how this result may be applied, in conjunction with a result of Barroero and Widmer on the enumeration of lattice points and o-minimal structures, to count real algebraic vectors occurring in families of semi-algebraic sets. This is used to determine the distribution of projective algebraic points via a suitable parametrization of projective space, which shall bring us towards a work in progress, in collaboration with Faustin Adiceam, on how one may analyse the Schmidt Subspace Theorem from the probabilistic perspective.

**Speaker:** Maiken Gravgaard

**Title:** Quantitative Khintchine on the parabola with non-monotonic approximation functions

**Abstract:** In metric Diophantine approximation Khintchine's Theorem states that if we take a function $\psi$ where the sum over natural numbers $q$ of $\psi(q)^n$ converges then almost no numbers in $\mathbb{R}^n$ are $\psi$-approximable. I will talk about a few natural ways to extend this theorem. First extending it to manifolds other than $\mathbb{R}^n$ and showing an effective version of this and then showing a strong version of this on the parabola in particular.

**Schedule:**

13:15- 14:00: Talk 1

14:00-14:10: Questions

14:10-14:40: Coffee, cake and fruit in Matkant

14:40-15:25: Talk 2

15:25-15:35: Questions

15:35-16:20: Talk 3

16:20-16:30: Questions

Later we will have a symposium dinner.

Organised by: ADA

Contact: Maiken Gravgaard
Revised: 15.04.2024