I had this question bothering me for a while, but I can't come up with a meaningful answer. 
The problem is the following:


Let integers $a\_i,b\_j\in${$1,\ldots,n$} and $K\_1,K\_2\in$ {$1,\ldots,K$}, then how small (as a function of $K$ and $n$), but strictly positive, can the following absolute difference be.


$\biggl|(\sum\_{i=1}^{K\_1} \frac{1}{a\_i})-(\sum\_{j=1}^{K\_2} \frac{1}{b\_j})\biggr|$


As an example for $K\_1=1,$$K\_2=1$ choosing $a\_1=n,$$b\_1=n-1$gives the smallest positive absolute difference, that is $\frac{1}{n(n-1)}$. What could the general case be? 



The difference of two sums of unit fractions
A sum of $K$ unit fractions, each of denominator $n\_i \leq n$, can 
be rewritten as a fraction with a denominator bounded by 
the product of the $n\_i$, i.e. by $n^K$. (A small improvement is possible, with product of distinct integers $\leq n$)


A difference of two such fractions  

(which are themselves sums of $K\_1$ and $K\_2$ fractions, respectively),
is a fraction with a demominator bounded by $n^{K\_1+K\_2}$.
(If $K\_1\neq K\_2$ the order of magnitude of the difference is actually
$\frac{1}{n^{\min(K\_1,K\_2)}}$.)


Let us assume that $K\_1=K\_2$.
So, the smallest nonzero difference $d(K,n)$ of sums of 
$K$ unit fractions, 
with $n\_i \leq n$ is $d(K, n)\leq \frac{1}{n^{2K}}$.


I believe that this order of magnitude, with slightly weaker constants,
 can be achieved with a constructive parametrization.


Example: If $K\_1=K\_2=2$, then choose
$\frac{1}{x^2 + 4 x + 1} + \frac{1}{x^2 + 4 x + 3} - 
\frac{1}{x^2 + 3 x + 1} - \frac{1}{x^2 + 5 x + 5}=
\frac{2}{(1 + 3 x + x^2) (1 + 4 x + x^2) (3 + 4 x + x^2) (5 + 5 x + x^2)}$.
Now, taking $n=x^2+5x+5$, one has a set of 4 unit fractions,
for which the difference of the sums above is asymptotically
$ \frac{2}{n^4}$. This example is (for $K\_1=K\_2=2$)
possibly the best one can find, (but I did not prove this).


I conjecture that for larger values of 
$K$ one can construct similar polynomial examples.


Quite possibly this has applications in questions in diophantine approximation,
exponential sums, large sieve etc. 


