Suppose we have a finite Egyptian fraction decomposition of a rational: $$\frac{n}{m} = \sum_{i=1}^k \frac{1}{x_i}$$ such that

(i) $x_i>0$,

(ii) $x_i \neq x_j$ for $i \neq j$, and

(iii) $\gcd(m, x_1,x_2,...x_k) = 1$.

Are their any known results concerning $\max_{i,j} |x_i-x_j|$ or maybe $\max_{i,j} |\frac{x_i}{x_j}|$?

For example,
$\frac{5}{121} = \frac{1}{26} + \frac{1}{350} + \frac{1}{275275}$ or $\frac{5}{121} = \frac{1}{33} + \frac{1}{93} + \frac{1}{3751}.$ Certainly the latter expression is "better" than the previous one for some vague notion of "better".

Note: Condition (iii) means we don't consider $\frac{5}{121}= \frac{1}{33}+\frac{1}{121}+\frac{1}{363}$ since this is really just a good decomposition of $\frac{5}{11}$ that has been divided through by $11$.

Motivation: I'm investigating a technique in my research that would produce Egyptian fraction representations where all the denominators are roughly the same size and I'm curious if this has been looked at before.


It's not totally clear from your question, but I assume you want $\max_{i,j} |x_i/x_j| = (\max_i x_i)/(\min_i x_i)$ to be as small as possible. (Ian's comment addresses the problem of making it large, for example.) It's also not clear whether you want to fix $k$ and consider this "smallest closeness measure" as a function of $k$, or whether you want the "smallest closeness measure" to be able to vary $k$ as well.

Note that you can never get $(\max_i x_i)/(\min_i x_i)$ to be smaller than about $e^{n/m}$, since the sum of the reciprocals of all the integers between $x$ and $e^{n/m}x$ is about $(\log(e^{n/m}x)+\gamma) - (\log x+\gamma) = n/m$. In fact, one can show that it's possible to get smaller than $e^{n/m}+\varepsilon$ for any given $\varepsilon>0$, as long as you allow $k$ to be large. See Croot's paper On unit fractions with denominators in short intervals, Acta Arith. 99 (2001), no. 2, 99–114.

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