We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and $\alpha $-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret’s tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.

A seminal result due to Wall states that if $x$ is normal to a given base $b$ , then so is $rx+s$ for any rational numbers $r,s$ with $r\neq 0$ . We show that a stronger result is true for normality with respect to the continued fraction expansion. In particular, suppose $a,b,c,d\in \mathbb{Z}$ with $ad-bc\neq 0$ . Then if $x$ is continued fraction normal, so is $(ax+b)/(cx+d)$ .

In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base- $b$ expansion. This partially answers a question of Bugeaud.

Let $g\,\ge \,2$ . A real number is said to be $g$ -normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\varphi$ denote Euler’s totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\varphi$ , $\sigma$ , or $\lambda$ , then the number

$$0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,.$$

obtained by concatenating the base $g$ digits of successive $f$ -values is $g$ -normal. We also prove the same result if the inputs 1,2,3....are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...

Let $g\geqslant 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\unicode[STIX]{x1D711}$ denote Euler’s totient function, let $\unicode[STIX]{x1D70E}$ be the sum-of-divisors function, and let $\unicode[STIX]{x1D706}$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\unicode[STIX]{x1D711}$, $\unicode[STIX]{x1D70E}$, or $\unicode[STIX]{x1D706}$, then the number

$$\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray}$$

$g$

$f$

$g$

$1,2,3,\ldots$

$2,3,5,\ldots$

$0.235711131719\cdots \,$