Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week.

**Talk Announcement:**

**Title:**Farey-subgraphs and Continued Fractions

**Speaker:**Seema Kushwaha (IIIT, Allahabad)

**When:**Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST

**Where**: Zoom: Please send email to the organisers for a link.

**Abstract.**

Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n}

\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.

\end{equation*}

The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

Let $\mathcal{X}$ be a subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a notion which is analogous to best rational approximation.

An element $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.

In this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties.

\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.

\end{equation*}

The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

Let $\mathcal{X}$ be a subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a notion which is analogous to best rational approximation.

An element $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.

In this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties.

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