K. Dajani

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math.DS2026

A strange continued fraction associated with the Romik map

Yufei Chen, Karma Dajani, Yanyan Hu +1

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed th…

math.DS2026

Coincidence of invariant measure for the alternate base transformations

Karma Dajani, Niels Langeveld

We characterize all pairs $(Î²,n),(Î²^\prime,m)$ such that the alternate $(I^^{2}, n)$ and $(Î²^\prime,m)$-transformations $K_{(I^^{2}, n)}$ and $K_{(Î²^\prime,m)}$ have the same absolutely…

math.DS2025

A unifying theory for metrical results on regular continued fraction convergents and mediants

Karma Dajani, Cor Kraaikamp, Slade Sanderson

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a sligh…

math.DS2025

Approximation properties of the intermediate $I^^{2}$ -expansions

Karma Dajani, Yan Huang

Given $I^^{2} > 1$ and $I^\pm \in [0, 1)$ , let $T_{I^^{2}, I^\pm} (x) = I^^{2} x + I^\pm (mod 1)$ . Then under the map $T_{I^^{2}, I^\pm}$ each $x \in [0, 1]$ has an \emph{intermediate $I^^{2}$ -expansion} of the form $x=\sum_{i=1…

math.DS2024

A new class of $I^\pm$ -Farey maps and an application to normal numbers

Karma Dajani, Cornelis Kraaikamp, Hitoshi Nakada +1

We define two types of the $I^\pm$ -Farey maps $F_{I^} \pm$ and $F_{I^\pm, ♭}$ for $0 < I^\pm < \frac{1}{2}$ , which were previously defined only for $\frac{1}{2} \leq I^\pm \leq 1$ by R.~Natsui (20…