On approximation by random L\"uroth expansions

TL;DR

This paper introduces a family of random L"uroth transformations that generate multiple L"uroth expansions for almost all points, analyzing their properties, convergence speeds, and digit frequencies, with implications for approximation quality.

Contribution

It develops a new class of random L"uroth maps, proving their ability to produce uncountably many expansions and analyzing their statistical and approximation properties.

Findings

For c ≤ 2/5, almost all x have uncountably many generalized L"uroth expansions.

For c=1/ell, almost all x have uncountably many universal expansions with digits ≤ ell.

The approximation speed for c=0 matches the standard L"uroth map, depending continuously on p.

Abstract

We introduce a family of random $c$-L\"uroth transformations $\{L_c\}_{c \in [0, \frac12]}$, obtained by randomly combining the standard and alternating L\"uroth maps with probabilities $p$ and $1-p$, $0 < p < 1$, both defined on the interval $[c,1]$. We prove that the pseudo-skew product map $L_c$ produces for each $c \le \frac25$ and for Lebesgue almost all $x \in [c,1]$ uncountably many different generalised L\"uroth expansions that can be investigated simultaneously. Moreover, for $c= \frac1{\ell}$, for $\ell \in \mathbb{N}_{\geq 3} \cup \{\infty\}$, Lebesgue almost all $x$ have uncountably many universal generalised L\"uroth expansions with digits less than or equal to $\ell$. For $c=0$ we show that typically the speed of convergence to an irrational number $x$, of the sequence of L\"uroth approximants generated by $L_0$, is equal to that of the standard L\"uroth approximants; and that the quality of the approximation coefficients depends on $p$ and varies continuously between the values for the alternating and the standard L\"uroth map. Furthermore, we show that for each $c \in \mathbb Q$ the map $L_c$ admits a Markov partition. For specific values of $c>0$, we compute the density of the stationary measure and we use it to study the typical speed of convergence of the approximants and the digit frequencies.