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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.

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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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