To determine if there exists an irrational number with decimal expansion containing only 0s and 1s and continued fraction entries bounded by 9, we analyze the properties of such numbers.

Key Observations:

1. Decimal Expansion Constraints: Let ( x = 0.d_1d_2d_3\ldots ) where each ( d_i \in {0,1} ). Since ( x ) is irrational, its decimal expansion is non-repeating and infinite. The ...

Yes. In fact there are uncountably many such numbers.

Let

• K01 be the “missing-digit” Cantor set of numbers in [0,1] whose (fractional) decimal expansion uses only the digits 0 and 1. This is the self-similar set generated by the similarities x ↦ x/10 and x ↦ (x+1)/10. It is compact, perfect, has Lebesgue measure 0, and Hausdorff dimension dim_H(K01) = log 2 / log 10 ≈ 0.30103. Its (Newhouse...

Here is essentially everything one needs to know:

1. Two Cantor‐type sets.
   A) Let
   C₁ = { x∈[0,1] : every decimal digit of x is 0 or 1 }.
   Then Hausdorff‐dimension (C₁)=log 2/log 10≈0.3010.
   B) Let
   C₂(9) = { x∈[0,1] : in the continued‐fraction expansion x=[0;a₁,a₂,…] one has all aᵢ≤9 }.
   It is well‐known (Good 1941, Hensley 1992, Jenkinson-Pollicott etc.) ...

<think> The question is whether there exists an irrational number whose decimal expansion consists only of the digits 0 and 1, and whose continued fraction has all partial quotients (entries) less than or equal to 9. The user provided an example with 23 digits where the max partial quotient is 9, but wonders if we can have larger such numbers or even arbitrary large expansions.

First, recall that...

This is a fascinating question about the relationship between decimal expansions and continued fractions. Let me analyze this problem systematically.

Understanding the Problem

We're looking for an irrational number whose:

1. Decimal expansion contains only digits 0 and 1
2. Continued fraction expansion has all entries ≤ 9

Key Observations

First, let's understand why this is challenging. ...

Yes – in fact there are continuum many of them, and one can do the same with any bound k≥1 in place of 9.

To see why, let us look at the two sets that are involved.

(1) The “0–1–decimal” Cantor set
C := {x∈[0,1] : in base 10 every digit of x is 0 or 1}.
It is closed, perfect, and has Hausdorff dimension
dim C = log 2 / log 10 ≈ 0.301.

(2) The set of numbers with bou...

Is there an irrational number containing only 0's and 1's with continued fraction entries less than 10?

This is an interesting question about the intersection of two special sets of real numbers.

Let me start by understanding what we're looking for:

1. An irrational number
2. Whose decimal expansion contains only 0's and 1's
3. Whose continued fraction has all entries less than 10

Analysis...

This is a fascinating and deep question that touches on the relationship between two different representations of real numbers: their decimal expansions and their continued fraction expansions. The relationship between these is notoriously complex.

The short answer to your question is yes, such an irrational number exists.

In fact, there are uncountably many of them. However, constructing on...

Yes – not only does such a number exist, there are in fact uncountably many of them.
Below we give two complementary arguments:

1. a “set–theoretic’’ argument that shows existence (and abundance) without producing an explicit example, and
2. an explicit step–by–step construction that lets you continue your 0–1 decimal expansion as far as you like while keeping every continued-fraction coeffic...