Irrational Number Approximations by Egyptian Fractions

I've written about Egyptian fractions before in a post titled The Greedy Algorithm on the 2nd August 2020. However, this post deals with ways of representing proper fractions, like 5/7, as Egyptian fractions. How do we approximate an irrational number such as \(\pi\) by Egyptian fractions?

The question arose because the number associated with my diurnal age today, 26997, has a property that qualifies for admission in OEIS A132556:

A132556

Egyptian fraction representation for the cube root of 82.

In the case of \(82^{1/3}\) one obvious way of achieving this is to first get the initial digits of its digital representation. We find that:$$82^{1/3} \approx 4.3444814857686119017$$This means that the proper fraction to convert into an Egyptian fraction is:$$\frac{3444814857686119017}{10000000000000000000}$$Taking this fraction and using this algorithm, we find that:$$82^{1/3} \approx 4+\frac{1}{3}+ \frac{1}{90}+ \frac{1}{26997} +  \frac{1}{5832713646}+ \dots$$These initial fractions are in agreement with the OEIS output but subsequent ones are not. Of course, while the Egyptian fractions were ideally suited to the purposes for which the Egyptians put them, they were never intended as approximations for irrational numbers of whose existence they were oblivious.

Egyptian fraction representations are not unique and especially so when dealing with approximations of irrational numbers. The Engels expansion is an alternative and thus we have:$$82^{1/3} \approx 4+ \frac{1}{3}+ \frac{1}{90}+ \frac{1}{27000}+ \frac{1}{233280000} + \dots$$The best rational approximations for irrational numbers remain the progressive approximations afforded by the number's continued fraction. In case of the cube root of 82 we have the following progressive approximations:$$\frac{9}{2},\frac{ 13}{3}, \frac{126}{29}, \frac{391}{90}, \frac{ 1299}{299}, \frac{ 121198}{27897}, \frac{ 486091}{111887},\frac{ 2065562}{475445}$$Figure 1 shows an interesting and relevant response to the Quora question: Is there any pattern in the Egyptian fraction representation of Pi?

Figure 1

Thue-Morse Constant

On June 20th 2020, I made a post titled Prouhet-Thue-Morse Sequence named for Eugène Prouhet, Axel Thue, and Marston Morse (the Prouhet reference is sometimes omitted). By the way, the Thue part is named after Axel Thue, whose name is pronounced as if it were spelled "Tü" where the ü sound is roughly as in the German word üben. It is incorrect to say "Too-ee" or "Too-eh". Thus sayeth N. J. A. Sloane, June 12th 2018, in his comments about OEIS A010060 that lists the members of the sequence. 

It is a most interesting sequence and my blog post covers it quite well and has links to three interesting YouTube videos. However, there is a so-called Thue-Morse constant that is the topic of this post. The sequence begins:

 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, ... 

If we concatenate these binary digits, we get a binary number:

 \(P=0.0110100110010110100101100..._2\) 

This number can be converted a decimal and is represented by the Greek letter \( \tau \):$$\tau=\sum_{n=0}^{\infty} \frac{t_i}{2^{i+1}}=0.4124540336401075977 \dots$$where \(t_i\) is the \(i^{th}\) element of the binary Thue-Morse sequence. The number has been shown to be transcendental.

Figure 1 provides two interesting expressions for the Thue-Morse constant (source):

Figure 1

I came across the constant by means of my diurnal age investigation, discovering that the number associated with my diurnal age (26547) was a member of OEIS A096394:

A096394

Engel expansion of Thue-Morse constant.

The sequence begins 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547, ... and the comments state that:$$ 0.4124540336 \dots = \frac{1}{3}+\frac{1}{3 \times 5}+\frac{1}{3 \times 5 \times 6}+\frac{1}{3 \times 5 \times 6 \times 9} + \dots$$I made a post about Engel Expansions way back on September 28th 2016.

If we take 0.412454033640107597783361368258455283089 as an approximation of \(\tau\) and plug this into the SageMathCell formula listed in this post, we do confirm that 26547 is a member. To generate further members of the sequence however, the number of decimal places to which \( \tau \) needs to be approximated must be increased. Here is a permalink to SageMathCell while the code is listed below (blue for input and red for output).

x=0.412454033640107597783361368258455283089

u=x

E=[1]

F=[]

product=1

sum=0

for i in [1..15]:

    a=ceil(1/u)

    u=u*a-1

    E.append(a)

    product=product*a

    sum+=1/product

    F.append(1/product)

print(E, "... this is the Engels expansion")

[1, 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547] ... this is the Engels expansion