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On the distribution of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions by Anderson, Jack; Boca, Florin P.; Cobeli, Cristian; Zaharescu, Alexandru is a scholarly article available to read on EtoBox.

What is On the distribution of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions about?

We consider the set ${\mathscr C}_Q$ of Farey fractions $d/b$ of order $Q$ with the property that there exists a matrix $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \operatorname{SL}(2,{\mathbb Z})$ of trace at most $Q$, with positive entries and $a\ge \max\{ b,c\}$. We prove that the elements of ${\mathscr C}_Q$ are asymptotically distributed with respect to the probability measure with density $(1/(1+x) -1/(2+x))/\log (4/3)$ and that $({\mathscr C}_Q)_Q$ has a limiting gap distribution as $Q\rightarrow \infty$.

Author

Anderson, Jack; Boca, Florin P.; Cobeli, Cristian; Zaharescu, Alexandru

Published

2025

Language

EN