# On absolutely normal and continued fraction normal numbers

## TL;DR

This paper constructs a real number that is normal in all integer bases and in its continued fraction expansion, with a detailed method for its computation and proof of its properties.

## Contribution

It provides a novel explicit construction of a number that is normal to all bases and continued fractions, combining metric theorems and interval refinements.

## Key findings

- Constructed a number normal to all bases and continued fractions.
- The first n digits can be computed in O(n^4) operations.
- Achieved control over interval lengths to ensure normality.

## Abstract

We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main diffculty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers bases.