1
$\begingroup$

Motivation

In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the following set of EFRs for a rational number $q$:

$$\mathsf{E}(q)=\left\{X\in[\mathbb{N}]^{\mathit{fin}}: q=\sum_{x\in X}{1\over x}\right\}.$$ Subsequently, he defines the function $$\mathscr{E}\colon q\mapsto \sum_{X\in \mathsf{E}(q)} \left( \prod_{x\in X}{1\over x}\right).$$

Let's call $\mathscr{E}(1)$ the Egyptian Schweber constant. Noah himself points out that a lower bound for this constant can be obtained quite easily. For instance, we have $\mathscr{E}(1) > \frac{37}{36}$, as $\mathsf{E}(1)$ contains both $\{1\}$ and $\{2,3,6 \}$.

I am interested in obtaining upper bounds for the Egyptian Schweber constant. Below, I'll outline what I've tried to obtain such a bound.

Aproach

My idea of solving this problem is to first use a bound on the amount of possible equation Egyptian fractions of length $n$. To that end, it might be somewhat easier to slightly tweak Schweber's definition to take into account the length of the EFRs: $$\mathsf{E}_{n}(q)=\left\{X_n := \{x_1 , x_2 , \dots , x_n \}: \sum_{k=1}^{n} \frac{1}{x_{k}} = q, \ 0 <x_{1}<x_{2}<\cdots<x_{n} \ \right\}.$$

Then, according to p.8 of the following paper by Ronald Graham we have $$ | \mathsf{E}_{n}(1) | < c_{0}^{(1+\epsilon)2^{n-1} }, $$ where $c_{0} \approx 1.264. $

Next, we need to obtain a good lower bound for the average size of the denominators of $ \mathsf{E}_{n}(1)$. By definition, it amounts to the sum of the denominators, divided by their amount. Let's call the average size of the denominators $AV(n)$, and the sum $S(n)$. We then have $$AV(n) = S(n)/n. $$

Using the aforementioned source, we also find that $$AV(n) > c_{0}^{-(1+\epsilon)2^{n-1} } \sum_{X_{n} \in \mathsf{E}_{n}(1)} \frac{1}{n} \sum_{x \in X_{n}} x .$$

Moreover, we then have the following upper bound for the Egyptian Schweber constant: $$ \mathscr{E}(1) < \sum_{n=1}^{\infty} c_{0}^{(1+\epsilon)2^{n-1}} \times \left( \frac{1}{AV(n)} \right)^{n} .$$

Unfortunately, I haven't been able to find any sources on the average size of the denominators of the EFRs. I did find that Sylvester's sequence can give Egyptian fractions of length $n$ with the largest denominators. Also, I found the following relevant master's thesis, but I don't think any bounds for the average size of the EFRs are given in there either.

Question

What is known about the asymptotics (especially lower bounds) of the average size - or the sum - of the denominators of Egyptian fraction representations of one?

Note: I've also asked this question on MSE

Improve this question
$\endgroup$
3
  • $\begingroup$ Bounds for individual denominators are discussed at mathoverflow.net/q/233336 $\endgroup$
    – Max Alekseyev
    Commented Oct 14, 2023 at 13:29
  • $\begingroup$ @MaxAlekseyev Thank you for the link. I am not entirely sure though how to apply your results to this problem -- they give an upper bound, right? Not a lower a bound? $\endgroup$
    – Max Lonysa Muller
    Commented Oct 14, 2023 at 14:14
  • $\begingroup$ Yes, there are only upper bounds as lower bounds for individual terms are not that interesting. E.g., in the case of $E_n(1)$ we have $x_i \geq i+1$ for $i<n$ and $x_n\geq n$. $\endgroup$
    – Max Alekseyev
    Commented Oct 14, 2023 at 15:49

0

Reset to default

You must log in to answer this question.